WATER HAMMER WITH COLUMN SEPARATION: A REVIEW OF RESEARCH IN THE TWENTIETH CENTURY
ANTON BERGANT Litostroj E.I. d.o.o., Litostrojska 40, 1000 Ljubljana, Slovenia;
[email protected] ANGUS R. SIMPSON School of Civil and Environmental Engineering, University of Adelaide, Adelaide, South Australia, 5005;
[email protected] ARRIS S. TIJSSELING Department of Mathematics and Computer Science, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands;
[email protected]
ABSTRACT Column separation refers to the breaking of liquid columns in fully filled pipelines. This may occur in a water hammer event when the pressure drops to the vapor pressure at specific locations such as closed ends, high points or knees (changes in pipe slope). A vapor cavity, driven by the inertia of the parting liquid columns, will start to grow. The cavity acts as a vacuum, a low-pressure point, retarding the liquid columns, which finally starts to diminish in size when the liquid columns change flow direction. The collision of two liquid columns, or of one liquid column with a closed end, moving towards the shrinking cavity, may cause a large and nearly instantaneous rise in pressure. The large pressure rise travels through the entire pipeline and forms a severe load for hydraulic machinery, individual pipes and supporting structures. The situation is even worse: in one waterhammer event many repetitions of cavity formation and collapse may occur. This report reviews water-hammer-induced column-separation from the discovery of the phenomenon in the late 19th century, the recognition of its danger in the 1930s, the development of numerical methods in the 1960s and 1970s, to the standard models used in commercial software packages in the late 20th century. A comprehensive survey of laboratory tests and field measurements is given. The review focuses on transient vaporous cavitation. Gaseous cavitation and steam-condensation are beyond the scope of the report. There are more than 300 references cited in this review report.
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TABLE OF CONTENTS
1
2
INTRODUCTION ________________________________________________________________ 4 1.1
Fluid Transients _________________________________________________________ 4
1.2
Scope of this Report ______________________________________________________ 4
1.3
Previous Reviews ________________________________________________________ 4
1.4
Cavitation _____________________________________________________________ 5
1.5
Column Sparation _______________________________________________________ 5
1.6
Mathematical Models _____________________________________________________ 5
1.7
Report Outline __________________________________________________________ 6
WATER HAMMER, COLUMN SEPARATION AND CAVITATION_____________________ 6 2.1
Water Hammer in a Historical Context _______________________________________ 7
2.2
First Observations of Sub-Atmospheric Pressures during Water Hammer Events _______ 7
2.3 Vaporous Cavitation _____________________________________________________ 9 2.3.1 Cavitation inception and tensile stress _______________________________________ 9 2.3.2 Local vapor cavities _____________________________________________________ 9 2.3.3 Intermediate vapor cavities ______________________________________________ 10 2.3.4 Distributed vaporous cavitation or two-phase (bubble) flow ______________________ 11
3
2.4
Gaseous Cavitation _____________________________________________________ 13
2.5
Severe Pressure Peaks Following Cavity Collapse ______________________________ 14
2.6
Severity of Cavitation / Cavitation Intensity / Measures of Cavitation _______________ 19
MATHEMATICAL MODELS AND NUMERICAL METHODS ________________________ 21 3.1
Assumptions ___________________________________________________________ 22
3.2
Water Hammer Equations ________________________________________________ 23
3.3
Discrete Single Cavity Models______________________________________________ 24
3.4 Discrete Multiple Cavity Models ____________________________________________ 27 3.4.1 The discrete vapor cavity model (DVCM) _______________________________________ 28 3.4.2 The discrete gas cavity model (DGCM) _________________________________________ 33 3.5
Shallow Water Flow or Separated Flow Models ________________________________ 36
3.6 Two Phase or Distributed Vaporous Cavitation Models __________________________ 38 3.6.1 Two-phase flow equations for vaporous cavitation region __________________________ 40 3.6.2 Shock equations for condensation of vaporous cavitation region_____________________ 42 3.7
Combined Models / Interface Models ________________________________________ 44
3.8 Other Models __________________________________________________________ 47 3.8.1 Jordan algebraic model _________________________________________________ 46 3.8.2 FEM _______________________________________________________________ 47 3.8.3 Other _______________________________________________________________ 47 3.9
A Comparison of Models _________________________________________________ 48
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3.10 Modeling of Friction _____________________________________________________ 49 3.11 State of the Art - The Recommended Models __________________________________ 50 3.11.1 Discrete cavity models: limitations ____________________________________________ 51 4
EXPERIMENTS IN THE LABORATORY AND MEASUREMENTS IN THE FIELD ______ 53 4.1
Photographs of Cavity Formation ___________________________________________ 53
4.2
Laboratory Experiments with Liquid Column Separation ________________________ 54
4.3
Gas Release ___________________________________________________________ 61
4.4
Fluid Structure Interaction (FSI) ___________________________________________ 61
4.5
Problems with Pressure Transducers ________________________________________ 62
4.6
Field Measurements _____________________________________________________ 62
5
CONCLUSIONS ________________________________________________________________ 64
6
ACKNOWLEDGEMENT ________________________________________________________ 66
7
REFERENCES _________________________________________________________________ 66
8
NOMENCLATURE _____________________________________________________________ 87
3
1
INTRODUCTION
1.1 Fluid Transients Pipes installed in water supply systems, irrigation networks, hydropower stations, nuclear power stations and industrial plants are required to convey liquid reliably, safely and economically. Modern hydraulic systems operate over a broad range of operating regimes. Any change of flow velocity in the system induces a change in pressure. The sudden shut-down of a pump or closure of a valve causes fluid transients which may involve large pressure variations, local cavity formation, distributed cavitation, hydraulic and structural vibrations and excessive mass oscillations. In particular, the occurrence of liquid column separation may have a significant impact on subsequent transients in the system. Large pressures with steep wave fronts may occur when vapor cavities collapse and the practical implications are therefore significant. As an outcome, fluid transients may lead to severe accidents (Jaeger 1948; Bonin 1960; Parmakian 1985; Kottmann 1989; De Almeida 1991). 1.2 Scope of this Report Cavitation is a broad field of research. This review is confined to the macroscopic aspects of transient vaporous cavitation and to the important case of column separation. The basis for this report is formed by the Ph.D. theses of the three authors (Simpson 1986; Bergant 1992; Tijsseling 1993). The literature reviews of these three theses have been combined and updated with additional contributions and with publications from the 1990s and early 2000s. The aim was to give a historical account, to summarize the state of the art and to have a list of references as complete as practically possible. 1.3 Previous Reviews Previous literature reviews on column separation have been published. A selected bibliography on column separation was presented by Jaeger et al. (1965). Martin (1973) provided a valuable summary of the historical development of many aspects of water hammer analysis and included an extensive bibliography. Wylie and Streeter in their three books (1967, 1978a and 1993) give account of previous work, including their own, on column separation. De Almeida (1987) reviewed the period 1978 to 1987. During the period between 1971 and 1991 an international Working Group of the International Association for Hydraulic Research (IAHR) carried out a major research effort with respect to column separation in industrial hydraulic systems. The scope of the work included topics on bubble nucleation and dynamics, two-phase flow, mathematical modeling of the basic physical phenomena, numerical simulation of transients in industrial hydraulic systems, instrumentation, and laboratory and prototype measurements. The Group
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focused its efforts on the investigation of transients in long pipelines (aqueducts) and power plant cooling water systems. One of the main aims of the Group was the development of computer codes and the validation of these codes against experimental results. The outcomes of the work by the Group, presented at nine international round-table meetings, were finally summarized in an extensive synthesis report (Fanelli 2000). Material from this publication has been included in this review. 1.4 Cavitation Microscopic aspects of cavitation, cavitation erosion, cavitation noise and advanced cavitation theory are not considered in this review. These aspects are covered elsewhere (Knapp et al. 1970; Hammitt 1980; Trevena 1987; Young 1989; Brennen 1995; Li 2000). Valuable information may also be obtained from Arndt's (1981) review. The presence of air in water may also have a significant impact on transients. Air presence is considered briefly in this review, but the focus is on systems with very little air as these situations will, in most circumstances, result in the highest pressures. 1.5 Column Separation Column separation can be compared with the breaking of a solid rod; Galilei already described this analogy (Rouse and Ince 1957, 1963, p. 57). The phenomenon can nicely be demonstrated in a simple toy apparatus (Wylie 1999). The general policy in hydraulic design involving liquid transients is to avoid liquid column separation, as described in the water hammer design criteria of Parmakian (1955). Lupton (1953) noted that cumulative high surge pressures are dangerous and recommended “that in order to ensure immunity from surge danger, vacuous separation should be prevented throughout the system concerned”. Martin (1973) pointed out that both in Europe and in the USA the design for column separation or the avoidance of cavitation was still subject of many debates. Even today there is reluctance by many designers to accept the occurrence of any column separation. Surge tanks, air chambers, or large motor rotating moments of inertia are suggested to avoid liquid column separation. Several authors concluded that if it cannot be avoided, then steps should be taken to minimize the impact when the liquid columns rejoin. For example, condenser cooling water systems are low-head systems where macro-cavitation cannot be avoided because of the topographic and operational characteristics of the plant. 1.6 Mathematical Models More than a century ago Joukowsky (1900) mathematically described many of the physical aspects of wave propagation in liquid systems. He also observed and explained column separation. Water hammer numerical models give physically accurate results, especially as to the first pressure rise, when the pressure is above the liquid vapor pressure (Jaeger 1977; Chaudhry 1979; Fox 1989; Wylie and Streeter 1993). The
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decay of the pressure trace is usually quicker in reality due to effects of free air, unsteady friction and structural vibration. Cavitation occurs when the pressure drops to the liquid vapor pressure and the onephase flow is transformed to two-phase flow. The classical water hammer equations are not valid in regions of cavitation. The first objective of modeling column separation is to predict the pressures that occur when large vapor cavities collapse. The second objective is to predict the timing of the events. A third objective might be to predict the structural response of pipes and supports. 1.7 Report Outline This report is divided into five parts. Following this introduction, water hammer and the different types of cavitation (including column separation) that may occur in pipelines are discussed in detail. The third section considers the popular mathematical models and their numerical implementation. Large pressure peaks or short-duration pulses due to vapor cavity collapse are also considered in this section. Experimental studies of column separation and field measurements are summarized in Section 4. Finally, conclusions are drawn in Section 5.
2
WATER HAMMER, COLUMN SEPARATION AND CAVITATION
Column separation can have a devastating effect on a pipeline system. A famous example is the accident at Oigawa hydropower station in 1950 in Japan (Bonin 1960). Three workers died. The plant was designed in the early 20th century. A fast valve-closure due to the draining of an oil control system during maintenance caused an extremely high-pressure water hammer wave that split the penstock open. The resultant release of water generated a low-pressure wave resulting in substantial column separation that caused crushing (pipe collapse) of a significant portion of the upstream pipeline due to the external atmospheric pressure. As an outcome, designing pipelines to withstand full atmospheric pressure on the outside when vapor pressure occurs on the inside has been standard practice for many years. Jaeger (1948) reviewed a number of the most serious accidents due to water hammer in pressure conduits. Many of the failures described were related to vibration, resonance and auto-oscillation. Two of the failures involved liquid column separation. In one case a governor caused a valve to open suddenly and thus produced a negative wave that resulted in a local liquid column separation at a break in the pipe profile. When the liquid columns rejoined, quite strong pressure rises caused cracking of a concrete section of the penstock. Kottmann (1989) described two accidents related to column separation in which two workers died. List et al. (1999) reported severe damage to the lining of a 7 km long, 0.6 m diameter, pump discharge pipeline, finally resulting in leaks. The cause was vapor-cavity collapse.
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2.1 Water Hammer in a Historical Context During the second half of the 19th century and the first quarter of the 20th century, the majority of the publications on water hammer came from Europe. The conception of the theory of surges can, amongst others, be traced to Ménabréa (1858, 1862), Michaud (1878), Von Kries (1883), Frizell (1898), Joukowsky (1900) and Allievi (1902, 1913). Joukowsky performed classic experiments in Moscow in 1897/1898 and proposed the law for instantaneous water hammer in a simple pipe system. This law states that the (piezometric) head rise ∆H resulting from a fast (Tc < 2L/a) closure of a valve, is given by:
∆H =
aV0 g
(1)
in which, a = pressure wave speed, V0 = initial flow velocity, g = gravitational acceleration, L = pipe length and Tc = valve closure time. The period of pipe, 2L/a, is defined as the return time for a water hammer wave to travel from a valve at one end of the pipeline to a reservoir at the other end, and back to the valve. The theoretical analyses performed independently by Joukowsky (1900) and Allievi (1902, 1913) formed the basis for classical water-hammer theory. Joukowsky’s work was translated by Simin in 1904. Allievi's work was not known generally outside Europe until Halmos made an English translation in 1925. Gibson (1908) presented one of the first important water hammer contributions in English. He considered the pressures in penstocks resulting from the gradual closure of turbine gates. 2.2 First Observations of Sub-Atmospheric Pressures during Water Hammer Events Joukowsky (1900, pp. 31-32) was the first to observe (see Fig. 1) and understand column separation. He explained the events in his experimental main-pipeline-gate system literally as follows (translation from German by the third author of this review): "Starting at the moment of closure of the gate the water in the pipe is continuously being stopped, whereby it is being compressed, the pipe expands and the pressure increases with ∆p. When this state travels with the celerity a up to the main, the latter transmits back along the pipe the main pressure (a little raised due to water hammer in the main itself) and a velocity of the water, which is directed in the direction of the main. This phase first passes the cabins II and III (measuring points), as a result of which the pressure in the indicators (gauges) in these cabins falls to the pressure of the main. When however the mentioned phase reaches the gate, this instantaneously causes, since the velocity of the water is directed away from the gate, a decrease of the pressure at the gate. If thereby the velocity V is so large, that according to the theory the reduced pressure would be negative, a break of the water mass will occur. The water column will be separated from the gate, ahead of which a small rarefied void develops. Similar separations can also form in other parts of a liquid column, the parts towards which the reduced pressure propagated." and "The
7
condition, that the water column is separated from the gate, prolongs the duration of the reduced pressure and makes the second impact stronger than the first, because it takes place with the velocity, at which the liquid column speeds into the rarefied void." See also Simin (1904, pp. 381-382).
Figure 1 Pressure record exhibiting column separation. Horizontal axis: time (each dot indicates half a second). Vertical axis: pressure (Joukowsky-pressure of 15.3 bar). The upper horizontal line is the static pressure and the lower horizontal line is the atmospheric pressure. (Joukowsky 1900, Fig. 17; also shown by Simin 1904, Fig. 13, and by Moshnin and Timofeeva 1965, Fig. 1) Gibson (1908) performed water hammer experiments with closure and opening of a downstream valve in a laboratory pipeline apparatus. He indicated that gas release in a part of the pipe section was initiated by a low-pressure (negative) wave. Strowger and Kerr (1926) indicated that a full column separation in the draft tube of a reaction water turbine could cause severe hydrodynamic loads following the turbine load rejection. Thorley (1976) attributed the first work on vapor cavities to Hogg and Traill (1926) and Langevin (1928). Mostowsky (1929) presented the first theoretical analysis of column separation in an explanation of his laboratory measurements. The first major United States involvement in the study of water hammer came in the form of a symposium on water hammer in 1933 in Chicago (Proceedings 1933). The ASME Committee on Water Hammer presented an extensive bibliography. The committee reviewed and summarized existing theory on water hammer and evaluated various methods employed, which included mainly arithmetic methods of solution. The symposium focused on water hammer in simple conduits, complex conduits, compound conduits, pump discharge conduits and surge tanks. A set of standardized nomenclature was developed at this symposium. Liquid column separation did not receive any mention in the committee report. A paper presented at the symposium by Billings et al. (1933) dealt with “parting of the water column” related to high-head penstock design. The authors noted that dangerous instantaneous pressure rises often originate in the upper portion of the penstock, when the liquid column parted and rejoined abruptly.
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2.3 Vaporous Cavitation Considering liquid transients in a pipeline system – there are two different types of flow regimes. The first is referred to as the water hammer regime (or “no cavitation” case) where the pressure is above the vapor pressure of the liquid. The second is the cavitation regime where the pressure is at the vapor pressure of the liquid. Two types of cavitation in pipelines are now distinguished. The magnitude of the void fraction is the basis for identifying the two types (Kranenburg 1974a). It is defined as the ratio of the volume of the vapor to the total volume of the liquid/vapor mixture and has also been referred to as the vapor fraction (Streeter 1983). The symbol α is used for void fraction following the nomenclature introduced by Wallis (1969). The void fraction depends on the magnitude of the velocity gradient in the cavitating flow. The two types of cavitation in pipelines are: (i) discrete vapor cavity or local liquid column separation (large α) and (ii) distributed vaporous cavitation or bubbly flow (small α). 2.3.1 Cavitation inception and tensile stress The liquid's vapor pressure is adopted as the cavitation inception pressure in numerical models for transient cavitation (Wylie and Streeter 1993). However, there are a number of reported experiments with cavitation inception pressures (negative peaks) much lower than the liquid vapor pressure (Lee et al. 1985; Takenaka 1987; Fan and Tijsseling 1992; Simpson and Bergant 1996). Washio et al. (1994) even observed traveling tensile waves (negative absolute pressure waves) in a dead end branch of a main oil pipe. The magnitude of the negative absolute pressure is the tensile strength of the liquid. It is governed by the flow conditions and the cavitational properties of liquid and pipe walls. Intense pre-pressurization of the liquid helps. Plesset (1969), Overton et al. (1984) and Trevena (1984, 1987) gave an in-depth treatment of the subject of tensile stresses in liquids. Recent results have been presented by Williams et al. (1999), Williams and Williams (2000) and Brown and Williams (2000). Tensile stress is a meta-stable condition for the liquid, which in a transient event should be described by non-equilibrium thermodynamics. Shinada and Kojima (1987) measured small negative absolute pressures (not negative peaks) in transient cavitation tests. They attributed this to the effect of surface tension, which can be important when the cavitation zone consists of many miniscule bubbles. 2.3.2 Local vapor cavities A large vapor cavity or liquid column separation is considered to have a local character (Kranenburg 1974a). The local void fraction is comparable with unity and the local velocity gradient is large. Examples include the formation of a cavity adjacent to a closed valve, below a reaction turbine runner (at the draft
9
tube inlet), at a high point in a pipeline, or at an intermediate location along the pipeline as the result of two crossing low-pressure waves (intermediate vapor cavity – see Section 2.3.3). After the descriptions of column separation by Joukowsky (1900) and Mostowsky (1929), it was LeConte (1937) who presented some of the first experimental results for a local liquid column separation at a rapidly closing valve at the downstream end of a pipeline. LeConte (1937) also proposed an analytical development based on rigid liquid-column analysis. A restitution coefficient was introduced to provide a match between the analytical and experimental results. This coefficient was proposed to change for each pipe, and even between waves in a single pipe, and therefore it was concluded to be difficult to estimate. 2.3.3 Intermediate vapor cavities Lupton (1953) introduced the possibility of the formation of an internal gap or intermediate vapor cavity, not located adjacent to a hydraulic device (valve, turbine, etc.) or at a high point. Moores (1953) initiated consideration of the topic in a discussion of Lupton's paper, when he posed a question as to what happened when surge or water hammer waves meet. Lupton's reply stated that: “if the meeting of the waves were negative, and their sum exceeded the absolute head H existing initially, a gap would be formed”. Lupton (1953) presented an example that exhibited the sequence of events leading to formation of an intermediate vapor cavity after instantaneous stoppage of a pumping system. The example illustrated that an intermediate cavity did not necessarily have to be located at a high point or change of slope of the pipeline. In a Master's thesis at The University of Melbourne, O'Neill (1959) investigated the occurrence of “intermediate” vapor cavities using the graphical method. He noted that most previous studies overlooked the formation of these intermediate cavities, which impose an internal boundary condition in the pipe. O'Neill (1959) presented a method to account for the formation of intermediate cavities. Experimental results for a simple reservoir-pipeline-valve system were also presented. In addition, visualization studies of growth and decay of intermediate cavities that used a high-speed movie camera were presented. Some of the analytical examples exhibited short-duration pressure pulses - due to cavity collapse - that exceed the initial pressure rise, Eq. (1), due to the valve closure. The experimental pressure trace records did not exhibit this short-duration pressure pulse behavior. Sharp (1960, 1965a, 1965b) continued O'Neill's (1959) work, and considered the growth and collapse of small vapor cavities produced by a rarefaction wave. An ideal spherical cavity was analyzed using force plus momentum principles (Sharp 1965a). Experimental results, which included high-speed photography of single intermediate cavities, were presented. Sharp proposed that another type of cavity also existed “during the first and succeeding rupture phases for an entirely different reason”. A succession of a small number of cavities at regular intervals was asserted to form moving away from the valve (Sharp 1965a,
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1977). Reversal of the transient flow in the pipeline caused these cavities to collapse and a series of regular pulses resulted when the liquid column rejoined. A modification of the graphical method of solution was proposed, which assumed secondary type cavities at equal intervals along the pipeline. This phenomenon appears to be similar to the occurrence of distributed vaporous cavitation in pipelines. Jordan (1961) investigated column separation in a pumping system after pump failure. He developed an analytical method for computation of exact locations of intermediate cavities along the pipeline. He asserted that the location of the intermediate cavity found by the standard Schnyder (1932) - Bergeron (1935) graphical method is approximate. Simpson (1986), Simpson and Wylie (1989), and Bergant and Simpson (1999a) showed clear experimental evidence of intermediate vapor cavity formation. Experimental evidence of short-duration pressure pulses was also presented. 2.3.4 Distributed vaporous cavitation or two-phase (bubble) flow Distributed vaporous cavitation is a region of two-phase flow consisting of both vapor and liquid. In contrast to local or intermediate vapor cavities, distributed vaporous cavitation occurs over an extended length of the pipe with the pressure at the vapor pressure of the liquid. The occurrence of distributed vaporous cavitation is illustrated in an example of an upward sloping pipe by Simpson (1986, p. 38) and Wylie and Streeter (1993, pp. 197-198). Once a low-pressure wave has passed along the pipe, the velocity varies from one end of the pipe such that the fluid is tearing apart and the void fraction is growing. The void fraction in distributed cavitation zones remains small (close to zero). The difference between a local liquid column separation and distributed cavitation in a pipeline was first described by Knapp (1937b) in a paper at the second water hammer symposium (Proceedings 1937) which expanded an earlier reference (Knapp 1937a). He presented an example in which a pressure drop was produced by a rupture of the pipe just below the shutoff valve. The negative wave traveled up the pipeline unchanged, provided the wave front did not intersect the “zero pressure line”. The liquid column, between this point of intersection and the reservoir, cavitated partially, but no liquid column separation occurred. Reflection of the water hammer wave from the reservoir brought the liquid column back to its original state without cavitation and produced a corresponding pressure rise. He concluded that “further investigations were necessary to clear up completely such water hammer conditions with cavitation”. Knapp (1939) further developed the concept of vaporous cavitation in a discussion of LeConte's (1937) paper. Knapp (1939) concluded that cases involving vaporous cavitation could not be solved by the graphical method. Knapp (1939) attributed many penstock failures in the upper portion of the penstock to the pressure rise that
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resulted from liquid column separation at a “knee” in the profile. Previously cited causes for these failures had been described as of “obscure origin”. De Haller and Bédué (1951) presented an analytical treatment of liquid column separation in a discharge pipe resulting from a low-pressure wave propagating upwards along a pipe. They suggested that cavities could occur along long sections of pipeline rather than forming immediately over the entire pipe cross section. Lupton (1953) presented a summary of the graphical method, with one section devoted to “separation of water columns”. The treatise considered water hammer problems in pumping systems. Lupton (1953) provided a description of the sequence of events associated with distributed vaporous cavitation. He distinguished between a localized liquid column separation and a region of distributed cavitation. The transmission of a negative wave along an upward sloping frictionless pipe was considered. The wave dropped the pressure to the vapor pressure of the liquid somewhere along the pipe. Lupton stated that further along the pipeline “the drop brought about in the velocity must diminish as the gradient is further ascended. In other words, the residual velocity will increase as the wave reaches higher portions of the main. The liquid column within such rising reaches will therefore become separated from the more slowly moving portion and will degenerate into a series of thin slugs of liquid interspersed with vacuous spaces (or more accurately, spaces at absolute pressure equal to the vapor pressure of the liquid plus the partial pressure of any released dissolved gases)”. Jordan (1965) investigated localized column separation and distributed vaporous cavitation in pumping systems with horizontal, upward and downward sloping pipe sections. He developed an analytical method for treatment of the distributed vaporous cavitation zones. Jordan (1965) studied the effect of hydraulic grade line (HGL) and pipe slope on the formation of distributed vaporous cavitation zones. There was a reasonable agreement between the computed results and results from measurements performed in a laboratory apparatus at Turboinstitute, Ljubljana, Slovenia. The apparatus comprised a fast-closing upstream valve and an upward sloping pipeline. A vaporous cavitation region may result from the passage of a negative wave through part of a pipeline, in which the static pressure decreases in the direction of wave propagation owing to friction or pipeline slope (Kranenburg 1974a). In contrast, if the static pressure increases in the direction of wave propagation a vaporous cavitation zone cannot occur (Zielke and Perko 1985; Simpson 1986; Simpson and Wylie 1989). See Figure 2. This type of cavitation may extend over long parts of the pipeline. The void fraction is usually much less than unity for small velocity gradients. A decrease in the pressure-change results in a smaller velocity-change as the water hammer wave propagates along the pipe. Thus fluid particles, as the wave
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passes, end up having a faster velocity than the "earlier" particles. This causes the liquid to pull apart when the negative wave drops the pressure to vapor pressure.
Figure 2(a) (left) Fast closing valve generating cavitation in (a) downward sloping pipe, (b) horizontal pipe, and (c) upward sloping pipe. (Zielke and Perko 1985, Fig. 3) Figure 2(b) (right) Cavity formation at a knee (change in pipe slope). (Zielke and Perko 1985, Fig. 4) Comprehensive experimental evidence of the occurrence of distributed vaporous cavitation was given by Simpson (1986), Simpson and Wylie (1989), Bergant (1992) and Bergant and Simpson (1999a). Simpson and Wylie (1989) gave an illustrative example showing the formation of a distributed vaporous cavitation zone. 2.4 Gaseous Cavitation Many papers, starting in the 1970s and early 1980s, have addressed the effects of dissolved gas and gas release on transients in pipelines (Swaffield (1969-1970, 1972a, 1972b); Enever (1972); Kranenburg (1974b); Wiggert and Sundquist (1979); Wylie (1980); Baasiri and Tullis (1983); Hadj-Taieb and Lili 13
(1998); Kessal and Amaouche (2001)). One of the main features of liquids is their capability of absorbing a certain amount of gas with which they come into contact through a free surface. In contrast to vapor release, which takes only a few microseconds, the time for gas release is in the order of several seconds. Gas absorption is slower than gas release (order of minutes) (Zielke et al. 1989). Gas release occurs in several types of hydraulic systems (cooling water systems, long pipelines with high points, oil pipelines, etc.). Dissolved gas is an important consideration in sewage water lines and aviation fuel lines. Gases come out of solution when the pressure drops in the pipeline. If a cavity forms, it may be assumed that released gas stays in the cavity and does not immediately redissolve following a rise in pressure. Kobori et al. (1955) and Pearsall (1965) showed that the presence of entrained air or free gas reduces the wave propagation velocity and accordingly the transient pressure variations. A significant limitation in the numerical models proposed in each of the above studies was the need to make rather arbitrary assumptions regarding the rate of release of gas. Dijkman and Vreugdenhil (1969) investigated theoretically the effect of dissolved gas on wave dispersion and pressure rise following column separation. 2.5 Severe Pressure Peaks Following Cavity Collapse Angus (1935) presented one of the first papers dealing with large short-duration pressure pulses due to the collapse of a cavity in a pipeline. In one example, the collision of a liquid column with a closed valve resulted in a short-duration pressure pulse of large magnitude. The duration of the pressure pulse was about one tenth of a 2L/a period (Angus 1937). Heath (1962), in a Master's thesis at Georgia Institute of Technology, investigated a local liquid column separation. Both incompressible (rigid-column) and weakly-compressible (elastic) analytical analyses were presented. The graphical method was used for the weakly-compressible analysis and exhibited shortduration pressure pulses due to cavity collapse. Short-duration pressure pulses due to cavity collapse were not apparent in the experimental results presented by Heath (1962). A short-duration (t < 2L/a) pressure pulse (Simpson 1986) due to cavity collapse is defined as a delayed step in the pressure-time trace superimposed on a pressure wave as shown in Figure 3. Both Simpson (1986) and Bergant (1992) described in detail the sequence of events leading to a short-duration pressure pulse. A rapid closure of the valve in a reservoir-horizontal pipe-valve system as shown in Figure 3(a) can be used to illustrate this phenomenon. The small effect of friction is neglected in this example. Instantaneous valve closure stops the liquid flow. The hydraulic grade line (HGL) along the pipe is initially constant at H0. The instantaneous head rise at the valve is predicted by the Joukowsky Eq. (1) as
∆H = −
a ∆V g
(2)
14
Water Hammer Model
H / H0
Discrete Cavity Model B
3.
t / (L / a) 8.
2.
B
6. AB
A
A
4.
1.
2. 0. 0.
2.
4.
6.
8. t / (L / a)
Hv / H0 -1.
0.
0.
1. x / L
b)
H0 V0
c) a)
Figure 3 A short-duration pressure pulse. (a) Reservoir-pipe-valve system. (b) Wave paths in distance-time plane. (c) Piezometric head history at valve. The pressure wave travels towards the reservoir, where it reflects negatively, and returns to the closed valve arriving at time 2L/a after the valve closure. Figure 3(b) shows the pattern of propagation and reflection of pressure waves in the system. In addition, Figure 3(c) shows pressure traces next to the valve that are predicted by a pure water hammer model (thin solid line) and by a single cavity model (thick solid line). The liquid flows in the reverse direction (towards the reservoir) after time 2L/a. Complete stoppage of the flow at the valve now requires a head drop of (a/g)V0. This pressure-head drop for the case shown in Figure 3(a) (thin solid line) would result in a head less than the vapor pressure-head (horizontal dashed line). When the pressure drops to the liquid vapor pressure (thick solid line in Fig. 3(c)), the velocity in the reverse direction is not reduced to zero when the water hammer wave reflects from the valve, but to (Mostowsky 1929)
V0 − ∆Vvc = V0 −
g ( H 0 + H b − pv* / γ ) a
(3)
where H0 = static HGL at the valve, Hb = barometric head,, pv* = absolute vapor pressure at temperature T, and γ = specific weight of liquid. Now, at the closed valve, the flow is still towards the reservoir. The liquid detaches from the valve and a cavity at the valve begins to grow. The cavity acts as a fixed-pressure boundary condition and water hammer waves with travel time L/a propagate to and from the reservoir. The
15
reverse liquid velocity decreases by ∆Vvc (given by Eq. (3)) each time a water hammer wave reflects off the vapor cavity. Once the velocity next to the valve becomes positive, the cavity begins to shrink until it finally collapses (point A in Fig. 3). The head directly caused by the cavity collapse is less than the waterhammer head generated at the beginning of the transient event, but the maximum pressure-head (greater than Joukowsky) occurs, in this example, at a time of about 6L/a in the form of a narrow short-duration pulse (point B in Fig. 3). This pulse is the superposition of the cavity-collapse head and the reservoir wave head doubled by its reflection from the closed valve. The resulting maximum head (thick solid line) is greater than the maximum head predicted by the Joukowsky equation (thin solid line). If the cavity collapse would occur exactly at the arrival of a pressure wave front, then a short-duration pulse would not occur. Wylie and Streeter (1993, pp. 192-196) also give a detailed description of the formation of a short-duration pressure pulse. They showed that the duration of existence of the first vapor cavity approximately is
Tcs =
V a 2L L ∆H 2 L 2a V0 = 0 = g∆H in a g∆H in a ∆H in a
(4)
where ∆Hin = H0 + Hb - pv*/γ is the head drop at which vaporization starts (cavitation inception head drop), thereby confirming the formula found by Mostowsky (1929) shown in Figure 6(a) herein. Walsh (1964) in a Master's Thesis at Syracuse University and Li and Walsh (1964) considered the prediction of the maximum pressure generated by the collapse of the first cavity. Photographs showed a cavity behind a closing valve. Minute bubbles probably indicated the presence of distributed vaporous cavitation. One-dimensional equations of motion for a weakly-compressible liquid were applied assuming a frictionless system. Laplace transforms were used to solve the system for velocity and head (Walsh 1964). A “linear throttling” case was considered in detail. The pressure generated by the collapse of the cavity was divided into two components: the first component was caused by the velocity of the liquid column at the valve just before the cavity collapsed. The second component was caused by the maximum pressure in the pipe at the instant just before the collapse of the cavity. The maximum head due to collapse of a cavity was given by
H max =
a V + 2 H RV g f
(5)
where Vf = velocity of the liquid column at the valve just before cavity collapse and HRV = difference in elevation between the downstream reservoir and the vapor pressure-head elevation at the valve. Similar equations were presented by Moshnin (1961) and Moshnin and Timofeeva (1965), Simpson and Wylie (1985) and Wylie and Streeter (1993, p. 194: Hmax = ∆H + 2∆Hin) for the case of instantaneous closure of a
16
downstream valve, where Hmax can be twice the Joukowsky value. The situation can even be worse when intermediate cavities form. Kottmann (1989) used rigid-column theory to show that the collapse of a midpoint cavity caused pressure rises of three times Joukowsky. For “non-linear throttling,” Li and Walsh (1964) stated that the maximum pressure in the pipe before collapse of the cavity was HRV, if the valve discharge monotonically decreased during the period from the time of valve closure to the beginning of liquid column separation. Eq. (5) also held for these conditions. Li and Walsh (1964) carried out experimental tests in a 2-inch plastic pipe to substantiate the analytical results. Walsh (1964) presented results for a number of experimental runs. Small steps in the pressure diagram or short-duration pressure pulses due to cavity collapse may be noted in a couple of the experimental results. The time of existence of the first cavity was quite long with respect to 2L/a. The velocity Vf was computed from experimental results using Eq. (5). Yamaguchi and Ichikawa (1976) and Yamaguchi at al. (1977) presented oscilloscope traces of experimental results exhibiting what appear to be the first results in the literature that clearly show short-duration pressure pulses. An upstream and a downstream closing valve were considered, with oil column separation in laminar flow being the focus of these studies. Photographs of cavity formation and collapse at a valve were also shown. Gottlieb et al. (1981) investigated maximum pressures following the collapse of cavities in a pipeline. They presented experimental results and numerical calculations. The numerical model was a simple discrete cavity model for each calculation point based on the method of Streeter and Wylie (1967). Four different configurations of a steel and a plastic pipeline were considered. Extremely high-pressure peaks were recorded immediately upon collapse of the vapor cavity. The pressure then dropped to about 40% of the pressure peak level and maintained this level for 2L/a seconds. The authors concluded the presence of peaks resembled the pressure peaks associated with the implosion of gas bubbles in pumps. See also Section 4.5 on the specific properties of pressure transducers. Martin (1983) measured transient cavitation in a simple reservoir-pipe-valve system. The water contained a minimal amount of dissolved gas. Limited cavitation (where the duration of the existence of the cavity at the valve is relatively short) was emphasized in contrast to previous studies (on severe cavitation) reported in the literature. The experimental results showed that the maximum pressure may exceed the Joukowsky pressure rise (Eq. (1)) in the form of a short-duration pressure pulse. See Figure 4 (top). Unfortunately, the reservoir pressure was rising during the experiment because the tank was too small. The measured reservoir-pressure shown in Figure 4 (bottom) was thought to be typical for all experimental runs.
17
Figure 4 Experimental result showing short-duration pressure pulses (Martin 1983, 1989). Absolute hydraulic grade line (HGL) variation with time at valve (top) and in reservoir (bottom). Graze and Horlacher (1983) investigated the possible occurrence of severe pressure peaks following cavity collapse. Previously reported severe pressure peaks at the beginning of the cavity-collapse pressure-rise were the focus of their study. Numerical results based on a discrete vapor cavity model with no air or gas evolution matched the experimental results reasonably well. The experimental runs performed in various pipe configurations had a time of existence of the first cavity of approximately 16 to 20 L/a. Short-duration pressure pulses were not evident in the results, probably due to the length of time of existence of the first cavity at the valve. The authors used both “piezo-resistive” and “inductive” types of transducers and found overshoot occurring with the inductive (strain-gage type) transducer, as exhibited by the measurement of pressures less than vapor pressure-head. Different inductive transducers produced different pressure peaks. In contrast, the piezo-resistive transducers did not produce these pressure peaks. The authors stated that “the pressures caused by the collapse of the vaporous cavity in the case of water column separation are of the same, or less, magnitude as those for the water hammer due to rapid valve closure”.
18
De Almeida (1983), in referring to experimental measurements by Van de Sande and Belde (1981) and Gottlieb et al. (1981), each of whom presented pressure peak values higher than those calculated by the Joukowsky formula, commented: “though old, this apparently very simple problem has not been resolved yet”. De Almeida (1983) cited five possible reasons for obtaining large pressures due to cavity collapse including: non-uniform velocity distribution, unsteady friction effects, “column elastic effect”, local and point effects. The “column elastic effect” was the result of the time of existence of the cavity not being an integer value of 2L/a. He presented an expression for estimating the upper bound of the overpressure in a frictionless system. Kojima et al. (1984) presented a mathematical model for predicting both the pressure rise associated with cavity collapse and the duration of the column separation. Photographs of cavity formation and collapse on the downstream side of a valve were presented. Their experimental results exhibited short-duration pressure pulses. The influence of unsteady laminar friction was studied. The authors concluded that the effects of energy dissipation due to non-adiabatic behavior of gas bubbles in a liquid column and the surface tension in the gas bubbles forming the separated cavity may be neglected. A “gas non-bubbly flow” model using unsteady pipe friction accurately predicted the experimental results. Simpson (1986) showed a range of short-duration pressure pulses measured in a reservoir - upward sloping pipeline - valve system. See the Figures 13, 18 and 19 herein. Due to the upward slope of the pipe the vapor cavity was confined to be adjacent to the valve with no distributed cavitation along the pipe (at least until the collapse of the first vapor cavity). 2.6 Severity of Cavitation / Cavitation Intensity / Measures of Cavitation The location and intensity of column separation is influenced by several system parameters including the cause of the transient regime (rapid valve-closure, pump failure, turbine load-rejection), layout of the piping system (pipeline dimensions, longitudinal profile and position of the valves) and hydraulic characteristics (steady flow-velocity, static pressure-head, skin friction, cavitational properties of the liquid and pipe walls) (Bergant 1992; Bergant and Simpson 1999a). The combination of several influential parameters creates difficulties in defining design criteria on cavitation severity. For a simple reservoir-pipeline-valve system the pressure rise following cavity collapse at the valve may or may not exceed the Joukowsky pressure rise Eq. (1) and cavities may form at the valve and/or along the pipe (Martin 1983; Simpson 1986; Carmona et al. 1987, 1988; Brunone and Greco 1990; Anderson et al. 1991; Simpson and Wylie 1991; Bergant 1992; Bergant and Simpson 1999a). Martin (1983) inferred the severity of cavitation from the duration, Tcs , of the first, mostly largest, column separation at the valve. He introduced the cavitation severity index S = Tcs a /(2L). He classified the intensity
19
of column separation as limited, moderate or severe cavitation with respect to the “number of bubbles” which may form in the pipeline. Paredes et al. (1987) defined the parameter indicating the 'severity of water column separation' as the ratio of the instantaneous pressure just before the first negative wave reaches the cavity boundary minus the instantaneous pressure that would be reached immediately afterwards, assuming that no column separation occurs, to the absolute instantaneous pressure just before the first negative wave reaches the downstream boundary. Carmona et al. (1987, 1988) and Anderson et al. (1991) proposed a somewhat different index and Fan and Tijsseling (1992) based a measure on the structural time scale of a vibrating closed pipe excited by external impact. Bergant and Simpson (1999a) performed a parametric numerical analysis on a reservoir-pipe-valve system to compute the critical flow conditions that classify different column-separation regimes according to the maximum pressure. A broad range of values was taken for the initial flow velocity (V0), the static upstream tank head (HT) and the pipe slope θ. The time scales L/a and Tc were constant. The numerical results (DGCM) shown in Figure 5 clearly show the influence of the initial velocity, static upstream tank head and (small) pipe slope on the maximum head at the valve. The maximum dimensionless head is presented as the ratio of maximum head rise (Hmax−H0)v at the valve (H0 = steady-state head at the valve) to the Joukowsky head rise aV0/g. Water hammer without column separation occurs for low initial flow velocities. The magnitude of the short-duration pressure pulse (see Section 2.5) is related to the magnitude of the reflection at the reservoir of the low-pressure wave and the intensity of cavitation along the pipeline. The maximum head for higher-velocity column-separation cases gradually decreases to the Joukowsky head. The maximum cavity volume occurs at the valve, whereas smaller cavities are formed along the pipe. The stronger attenuation of short-duration pressure pulses in a downward sloping pipe may be attributed to the more intense cavitation along the pipe.
3.0 HT,1=5. m HT,1=10. m HT,1=15. m HT,1=20. m HT,1=25. m
2.5 2.0 1.5 1.0
(Hmax-H0)v / (aV0/g) (-)
(Hmax-H0)v / (aV0/g) (-)
3.0
o
θ = -3.2
0.5 0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40 1.60 (a)
V0 (m/s)
(b)
HT,2=7. m HT,2=12. m HT,2=17. m HT,2=22. m HT,2=27. m
2.5 2.0 1.5 1.0
o
θ = +3.2 0.5 0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40 1.60
V0 (m/s)
Figure 5 Computed maximum head at valve as function of initial velocity (adapted from Bergant and Simpson 1999a). (a) downward sloping pipe, (b) upward sloping pipe.
20
Bergant and Simpson (1999a) classified transient regimes according to the physical state of the liquid and the maximum pipeline pressure as: (1) Water hammer flow regime. No column separation occurs during transients. Joukowsky pressure. (2) Active column separation flow regime. The maximum pipeline pressure is generated following the first column separation at the valve and along the pipeline (active column separation from the designer’s point of view). The maximum pressure at the valve is governed by the intensity of the short-duration pressure pulse. Joukowsky pressure exceeded. (3) Passive column separation flow regime. The maximum pipeline pressure is the water hammer pressure before intense cavitation occurs. Joukowsky pressure. This classification is based on maximum pressures. The steepness of wave fronts is another important parameter in the assessment of dynamic loads on pipe systems and their supporting structures. Columnseparation collapse probably is the most important source of very steep wave fronts. For water hammer and column separation events, Fanelli (2000) has classified the following regimes: (1) no cavitation, (2) cavitation, but cavity does not reclose, (3) cavitation with cavity reclosure, but no high pressures, (4) cavitation with cavity reclosure, but with excessive pressures.
3
MATHEMATICAL MODELS AND NUMERICAL METHODS
Until the 1960s, a comprehensive investigation of liquid column separation and cavitation in pipelines was not possible due to the unavailability of computers. In a report on the status of liquid transients in Western Europe, Martin (1973) stated that “only in the last decade there has been concerted activity on transient problems for which vapor or gas was present in the liquid”. Until around 1960, most studies used graphical and arithmetic procedures originally set forth by Gibson (1919-1920), Schnyder (1932) (see Hager (2001)), Angus (1935), Stepanoff (1949), Bergeron (1950) and Parmakian (1955). The first computer-oriented procedures for the treatment and analysis of water hammer include work by Thibessard (1961), Lai (1961), Streeter and Lai (1962), Streeter (1963, 1964, 1965-1966), Van De Riet (1964), Vreugdenhil (1964) and Contractor (1965). In a preface to the proceedings of the Third Symposium on Water Hammer - Pumped Storage Projects (Proceedings 1965), Streeter (1965-1966) observed that the development of digital computer methods of water hammer analysis had occurred during the previous 5 years.
21
Some of the first studies of liquid column separation using a numerical computer simulation model include the work by Thibessard (1961) at Liège in Belgium, Streeter and Wylie (1967), Baltzer (1967a, 1967b) at the University of Michigan, Siemons (1967) at Delft in The Netherlands, and Weyler (1969) and Weyler et al. (1971) at the University of Michigan. Column separation (formation and collapse of large discrete cavities, evolution and condensation of vaporous cavitation regions) may be described by a set of one-dimensional equations each representing a particular physical state of the liquid. These include the states of liquid, a mixture of liquid and vapor bubbles, and discrete vapor cavities (Bergant and Simpson 1997, 1999a). A negligible amount of free or released gas in the liquid is assumed (Blake 1986; Hansson et al. 1982; Wylie 1984). The set of equations include (1) Water hammer equations describing the liquid state, (2) Equations for a discrete vapor cavity or an intermediate cavity that separates liquid and vaporous cavitation regions along the pipeline or at boundaries (e.g. valve, knee), (3) Two-phase flow equations for a homogeneous mixture of liquid and liquid-vapor, (4) Shock equations for the condensation of liquid-vapor mixtures back to the liquid phase. Each of these various types of flow regime is described in the subsequent sections. The derived equations may be applied to all types of pipe configurations and to various cases of phase interactions. 3.1 Assumptions There are a number of assumptions made in relation to the development of mathematical models (Bergant and Simpson 1997, Bergant and Tijsseling 2001, Bergant et al. 2003). The assumption of one-dimensional flow is made including average cross-sectional values of flow velocity, pressure, density and void fraction. Only the assumptions related to column separation are detailed here: (1) All liquids have entrained free gases or dissolved gases that evolve when the pressure drops below saturation pressure. This review mainly considers flow situations where the presence of substantial quantities of air is not considered. Air release is considered in detail by Wylie and Streeter (1993, pp. 178-184). Vaporous cavitation is inertia-driven cavitation and sub-cooled (the equations are valid up to a temperature of about 330 K, above which thermodynamic effects become important (Hatwin et al. 1970)). (2) The volume of the vapor cavity must be significantly less than the volume of a reach in the numerical model. The pressure in the cavity is equal to the vapor pressure. Water hammer waves are reflected off the
22
cavity, which is assumed to occupy the total cross-sectional area of the pipe. The vapor cavity does not move. Mass and momentum of the vapor in the cavity are negligible. Isothermal conditions in the cavity prevail. The vapor condenses completely prior to the instant of rejoining of liquid columns or cavity collapse against the boundary. The formation of cavities has no effect on friction losses in the pipeline. (3) The void fraction of the vapor in a mixture of liquid and vapor bubbles (distributed vaporous cavitation zone) is much smaller than unity, so the mass and momentum of the bubbles can be neglected. The surface tension effect that results in a pressure difference across the vapor bubbles is ignored (Sundquist 1977). The liquid and the vapor-bubble velocities in the mixture are the same during vaporous cavitation (Prosperetti and Van Wijngaarden 1976). The vapor bubble is not influenced by the expansion and compression of the neighboring liquid-vapor bubbles (Brennen 1973). The influence of gravity on the bubbles is neglected. (4) There is an infinitesimal width of discontinuity between the interface (shock wave front) of the onephase fluid (liquid) and the one-component two-phase fluid (homogeneous mixture of liquid and vapor bubbles). Increase in temperature across a shock wave front is small and therefore isothermal conditions across the interface prevail (Campbell and Pitcher 1958). 3.2 Water Hammer Equations The water hammer equations are applied to calculate unsteady pipe liquid flow when the pressure is (always and everywhere) greater than the vapor pressure. They comprise the continuity equation and the equation of motion (Wylie and Streeter 1993): ∂H ∂H a 2 ∂V + V − V sinθ + = 0 ∂t ∂x g ∂x
(6)
f V |V | ∂H ∂V ∂V + + V + = 0 ∂x ∂t ∂x 2D
(7)
g
where H = piezometric head (hydraulic grade line HGL), t = time, V = flow velocity, x = distance along the pipeline, θ = pipe slope, a = pressure wave speed, g = gravitational acceleration, f = Darcy-Weisbach friction factor (its history is nicely described by Brown 2002) and D = pipe diameter. For most engineering applications, the convective terms V(∂H/∂x), V(∂V/∂x) and Vsinθ are very small compared to the other terms and therefore neglected. Research by Streeter and Wylie (1967) led the world to the direct use of the method of characteristics as a numerical method on a digital computer to provide solutions to Eqs. (6) and (7). The method
23
of characteristics has been the standard solution method for solving water hammer in pipeline systems for the last 40 years. 3.3 Discrete Single Cavity Models Single vapor cavity numerical models deal with local column separations as described in Section 2.3.2. A single cavity is used either at a boundary, at a high point in the pipeline, or at a change in pipe slope. Most graphical solutions of water hammer problems employed this modeling approach. Rigid column theory has also been used to compute the behavior of systems with single cavities.
Figure 6(a) Theoretical analysis of column separation: pressure as function of time (Mostowsky 1929, Fig. 7).
Figure 6(b) (left) Theoretical analysis of column separation: velocity as function of time (Mostowsky 1929, Fig. 8). Figure 6(c) (right) Laboratory measurement of column separation: pressure as function of time (Mostowsky 1929, Fig. 16).
24
Mostowsky (1929) analyzed column separation at a downstream valve. Figure 6(a) shows the pressure-time diagram for a frictionless pipe where the duration Tcs of the column separation is an integer multiple of the wave travel time 2L/a. This figure is a confirmation of Eq. (4). Figure 6(b) is the corresponding velocitytime diagram that determines the time of separation. Mostowsky performed measurements in a 29.5 m long, 2 inch diameter pipe. The measured pressure history in Figure 6(c) shows that, unlike Figure 6(a), the second pressure rise is lower than the first one, and the measured number of 11-times 2L/a is not the 12times predicted by Eq. (4). Mostowsky, attributing these discrepancies to friction, performed a rigid column analysis with a quadratic friction law. He developed closed-form solutions pertaining to rigid-column theory, for example a formula relating the effective friction coefficient to P1 and P2 (these are indicated in Fig. 6(c)), and an improved formula for Tcs giving a value of 10.6-times 2L/a in agreement with the experiment. Angus (1935) presented one of the first mathematical models of a single vapor cavity at a boundary. A graphical method was presented (Fig. 7). For a pump failure on a discharge line, the possibility was investigated of a cavity forming on the pipe side of a check valve near the pump. After the cavity formed and the liquid column returned to the closed valve, the pressure rise was found to be about “four times” the normal value. This was some of the first literature dealing with large short-duration pressure pulses due to the collapse of a cavity in a pipeline. Angus (1937) prepared a bulletin on water hammer that included an example of a power failure for a pump with a small inertia. The example assumed a pump discharge valve closed instantly when the pump failed. A vapor cavity formed on the pipe side of the valve and existed for about 2L/a seconds. The eventual reversal of the liquid column resulted in a short-duration pressure pulse of large magnitude after the cavity collapsed. The duration of the pressure pulse was about one tenth of a 2L/a period.
Figure 7 Graphical method by Angus (1935, Fig. 6).
25
In a discussion of LeConte's paper (1937), Bergeron (1939) presented an example of the formation of a cavity at a valve. The graphical method was used which included "lumped" friction losses. Bergeron (1950) published a book that provided an extensive treatment of water hammer in hydraulics and surges in electricity generation with the English translation being published in 1961. Treatment of a local liquid column separation was similar to that of Angus (1935), however, examples were presented in which the cavity existed for longer than 2L/a seconds. Improvements were made in the schematization for local liquid column separations. Bergeron indicated that the underpressure in a cavity was not the “barometric vacuum”, but rather the vapor pressure at the temperature of the liquid. He neglected the influence of gravity on the shape of the liquid-vapor interface. The interface was assumed to be perpendicular to the pipe and the pressure at the local liquid column separation was equated to the vapor pressure of the liquid. In one example, a cavity − formed at the valve after 2L/a seconds − collapsed at an exact multiple of 2L/a seconds; in other words, just as the initial wave due to the instantaneous valve closure arrived back at the valve. In another example conditions were selected such that the cavity collapsed at a time that was not a multiple of 2L/a seconds after the valve closed. The collapse of the cavity generated a second waterhammer wave in the system with a different timing than the initial water-hammer wave created by the closure of the valve. When the initial valve-closure wave returned to the valve for the first time after collapse of the cavity, a distinct step in the pressure-time trace was generated. The maximum pressure of the step did not exceed the magnitude of the initial Joukowsky pressure rise due to the valve closure. In this application of the graphical method, friction was lumped at the upstream end by using an orifice. A number of cycles of successive formation and collapse of a local liquid column separation was considered. Friction loss resulted in damping and the mean value of head rise decreased with every cycle, as did the maximum volume of the cavity. Kephart and Davis (1961) used rigid-column theory to determine the magnitude of the pressure rise when liquid columns rejoined in pump discharge lines equipped with check valves at the pump outlet. This method served as a preliminary design technique. Escande (1962) included cavities and lumped non-linear friction in his calculations. Streeter and Wylie (1967) presented a computer model describing vaporous cavitation using only a single vapor cavity in a pipeline. Rupture of a pipeline just below an upstream reservoir was considered. A single cavity was assumed to form at the point in the pipeline that first dropped to the vapor pressure of the liquid. Weyler (1969) continued studies of liquid column separation at The University of Michigan. His numerical model used a single vapor cavity at the valve. Cavities were not permitted to form at the other computational sections. A semi-empirical “bubble shear stress” was proposed to predict the increased momentum loss observed under liquid column separation conditions. The “bubble shear stress” was
26
attributed to the non-adiabatic expansion and collapse of gas bubbles present throughout the low-pressure flow. Microscopic bubbles of permanent gas, usually air, served as nuclei for the explosive vaporous growth. More air was picked up by diffusion of dissolved air in the liquid during the initial phase of the transient. Air remained undissolved accounting for the decrease in wave speed. A single spherical bubble was examined and compressive dissipation was concluded to be large compared with the viscous dissipation for a single spherical bubble. De-aerated liquid was found to undergo much more violent opening and closing of cavities, a behavior characteristic of distributed vaporous cavitation. Safwat (1972b) also considered the wave attenuation problem and introduced an equivalent shear stress concept. Kranenburg (1974a) disagreed with Weyler and Safwat and contended that the thermodynamic behavior was essentially isothermal (because of the small size of the bubbles) and that no dissipation would occur due to this bubble shear stress mechanism. The growth and subsequent decay of the discrete vapor cavity is calculated by the continuity equation: t
∀vc =
∫ A (V − V ) d t
(8)
u
tin
where ∀vc = discrete vapor cavity volume, tin = time of inception of the discrete vapor cavity, A = crosssectional pipe area, V = outflow velocity at the downstream side of the vapor cavity, and Vu = inflow velocity at the upstream side of the vapor cavity. 3.4 Discrete Multiple Cavity Models This type of model includes the concentrated or discrete vapor cavity models (DVCM). Liquid column separations and regions of vaporous cavitation are both modeled using discrete cavities at all computational sections. Liquid is assumed to be in between all computational sections and the method of characteristics is applied throughout the pipeline, even in vaporous cavitation regions. The discrete free gas cavity model (DGCM) (Wylie 1984, Zielke and Perko 1985) is similar to the discrete vapor cavity model, with a quantity of free air assumed to be concentrated at each computational section. The only difference with DVCM is in the ∀ - p* curve as shown in Figure 8. Bergeron (1950) considered the formation of a cavity at a location away from a pump. Potential cavity locations were identified from the profile of the pipeline. The cavity at the valve was treated correctly. However, O'Neill (1959) and Sharp (1965a, 1965b) pointed out that Bergeron's work did not account for an intermediate vapor cavity that formed away from the control valve, following the first collapse of the cavity at the valve. The meeting of two low-pressure water-hammer waves produced a potential pressure considerably less than vapor pressure and resulted in the formation of an intermediate cavity.
27
Figure 8 p*-∀ curves for DVCM (thick solid line, ∀0 = 0 mm3) and DGCM (thin solid lines, ∀0 = 0.01 mm3, ∀0 = 0.1 mm3, ∀0 = 1 mm3, ∀0 = 5 mm3, ∀0 = 10 mm3) with p0* = 1 bar and pv* = 0.2 bar. The value of pv* (water at 60 oC) is chosen large to show its influence.
3.4.1
The discrete vapor cavity model (DVCM)
The discrete vapor cavity model (DVCM) continues to be the most commonly used model for column separation and distributed cavitation at the present time. One significant advantage of the DVCM is that it is easily implemented and that it reproduces many features of the physical events of column separation in pipelines. Bergeron (1950, pp. 89-95) and Streeter and Wylie (1967, p. 209), described in general terms a procedure for calculating liquid column separation and rejoining (single cavity). The first development of the discrete vapor cavity model (multiple cavities) appears to be by Thibessard (1961) and independently by Streeter (1969) and Tanahashi and Kasahara (1969).
Figure 9 Definition sketch for discrete cavity model (Tijsseling et al. 1996, Fig. 8)
28
Wylie and Streeter (1978a, 1993) have described the DVCM in detail and they provided a FORTRAN computer code for its implementation. Cavities are allowed to form at any of the computational sections if the pressure is computed to become below the vapor pressure of the liquid. The DVCM does not specifically differentiate between localized vapor cavities and distributed vaporous cavitation (Simpson and Wylie 1989, Bergant and Simpson 1999a). Vapor cavities are thus confined to computational sections, and a constant pressure wave speed is assumed for the liquid between computational sections. See Figure 9. Upon formation of a cavity, a computational section is treated as a fixed internal boundary condition. The pressure is set equal to the vapor pressure of the liquid until the cavity collapses. Both upstream and downstream discharges for the computational section are computed, using the C+ and C– compatibility relationships for each of the positive and negative characteristics within the method of characteristics (MOC) solution. These are (see Fig. 10 for the MOC x-t staggered grid): H P = C pc − B pc QPu
(9)
H P = C mc + Bmc Q P
(10)
where Cpc and Bpc are constants for the positive or plus characteristic (pc), and Cmc and Bmc are constants for the negative or minus characteristic (mc) based on HP and QP at the previous time step at adjacent computational sections A and B in Fig. 10.
∆x ∆t
t
P
t+∆t C+ t
C-
A
B x
0.
1
L
2
j
N
N+1
Figure 10 The method of characteristics staggered grid in reservoir-pipe-valve system.
29
The vapor cavity volume change ∆∀vc at a computational section in terms of the difference in discharge at the downstream (QP) and upstream (QPu) side of the section is t +∆ t
∆∀ vc =
∫ (Q
P
− QPu ) d t
(11)
t
The volume of the cavity is cumulated at each computational section where vapor pressure exists. If the cavity volume becomes zero or negative, the computation returns to the standard method of characteristics (water hammer solution). Provoost (1976) used a closure condition that exactly satisfies the mass balance. Wylie and Streeter (1993, p. 69) showed a comparison of the results of the DVCM and the experimental results by Li and Walsh (1964) of an isolated cavity formation at the downstream side of a valve. The magnitudes of the pressures were reasonably predicted, whereas the timing of existence of column separation was not well predicted. Although the discrete vapor cavity model was correctly formulated by Thibessard (1961), Streeter (1969) and Tanahashi and Kasahara (1969), these investigators did not apply the model to regions of distributed cavitation. This was done by De Vries (1972, 1973, 1974) when he simulated the experiments at Delft Hydraulics Laboratory performed by Vreugdenhil et al. (1972) and Kloosterman et al. (1973). De Vries (1972) was the first to report on numerical oscillations induced by the simulated annihilation of a region of distributed cavitation. See Figure 11. To suppress these oscillations he added small amounts of free gas to the discrete cavities. In fact he then used the discrete (concentrated, lumped) free gas cavity model developed by Brown (1968). This model, described in Section 3.4.2 and used by Enever (1972), Tullis et al. (1976), Ewing (1980) and Suda (1990), allowed for the presence of free gas in the liquid. Ewing (1980) also discussed various damping mechanisms in liquid-gas mixtures. Evans and Sage (1983) had confidence in the DVCM and used it for the water-hammer analysis of a practical situation. Capozza (1986) applied the method to condenser cooling circuits. Wylie (1984) and Zielke and Perko (1985) gave thorough treatises of both the vapor and the free gas discrete cavity models. Bergant and Simpson (1999b) incorporated cavitation inception with “negative” pressure into the discrete vapor cavity (DVCM) numerical model. Numerical and measured results of experimental runs with “negative” pressure spikes were compared. The local "negative" pressure spike at cavitation inception did not significantly affect the column separation phenomena.
30
Figure 11 Numerical oscillations in DVCM results by De Vries (1972, Fig. 42). Pressurehead histories at different positions after pump shut-down. Bergant (1992) classified methods to attenuate unrealistic pressure spikes due to multi-cavity collapse as follows: (1)
Interpolation within the MOC grid (Numerical damping). Kot and Youngdahl (1978a) used spatial interpolation for a complete (with convective terms) set of water hammer compatibility equations. A similar approach has been adopted by Anderson and Arfaie (1991). The authors introduced a variable length of the liquid column due to formation, growth and collapse of vapor cavities along the pipe. Miwa et al. (1990) artificially corrected the numerical time step to ∆t = 0.95 ∆x / a and then used spatial interpolations. Spatial interpolation causes numerical damping (Goldberg and Wylie 1983), which may suppress physical pressure pulses as observed in measurements.
(2)
Additional damping mechanisms (Physical damping). Weyler et al. (1971) attributed additional (apparent) 'bubble shear stress' to the non-adiabatic expansion and collapse of gas bubbles present throughout the low-pressure cavitating flow. Similar models were developed by Safwat (1975) and De Bernardinis et al. (1975). Alexandrou and Wylie (1986) proved analytically that during column separation the 'thermodynamic' losses are much smaller than the losses due to skin friction at the pipe wall. Kojima et al. (1984) implemented Zielke's (1968) unsteady friction model into a column separation model. The authors investigated an oil-hydraulic system with transient laminar flow. They compared computational results with results of measurements and found that the consideration of unsteady friction in DVCM improved the numerical results. Similar approaches using unsteady friction model have been used by Shuy and Apelt (1983), Brunone and Greco (1990), Bergant and Simpson (1994a), Bughazem and Anderson (1996, 2000) and Bergant and Tijsseling (2001). Implementation of unsteady friction models into DVCM is discussed in Section 3.10.
(3)
Accounting for the discrete vapor cavity volume (Numerical damping). Safwat and Van Den Polder (1973) allowed discrete cavities to form only at predetermined locations like boundaries
31
and thus avoided problems with multi-cavity collapse. When the pressure at the internal computational section dropped below the vapor pressure, the discharge at this section was taken the average of the two discharges calculated from Eqs. (9) and (10), with HP set to Hv. Carmona et al. (1988) adopted a similar approach. A disadvantage of this method is that the user must preselect the potential locations where localized vapor cavities may form. Also, the averaging of discharges has no physical meaning. Golia and Greco (1990) allowed cavities to form at a selected number of internal pipeline computational sections. At other sections the pressure was allowed to drop below the liquid vapor pressure, which is physically not realistic. Several authors tried to control the numerical spikes with an adequate integration of Eq. (11). A generalized numerical integration scheme for the continuity equation describing the vapor cavity volume is (adopted from Wylie (1984) for the MOC with rectangular grid): t ∀tv = ∀tv−∆t + {ψ (QPt − QPt u ) + (1 −ψ ) (QPt −∆t − QPt −∆ u )} ∆t
(12)
where ∀vt is the vapor cavity volume at time t, ψ is a weighting factor, and QPut and QP t are the upstream and downstream discharges at the computational section at time t. The standard form of (12) is the trapezoidal rule which uses a ψ of 0.5 (Tanahashi and Kasahara 1969, 1970; Streeter 1972; Wylie and Streeter 1978a; Safwat et al. 1986; Liou 1999). Streeter (1969), Provoost (1976) and Bergant and Simpson (1999a) used a ψ value of 1.0. Bergant (1992) and Simpson and Bergant (1994a) performed numerical computations taking ψ values in the range from 0.5 to 1.0 and found that when using a ψ close to 1.0, which was also recommended by Liou (2000), that the pressure spikes were suppressed significantly. (4)
Allowing free gas in a cavity. (Physical damping). As stated in one of the previous paragraphs it was De Vries (1973) who first employed a discrete gas cavity model (DGCM) to suppress spurious pressure spikes and this model has since been used by several researchers. The DGCM and its properties are discussed in the next section.
(5)
Filters. (Numerical damping). Kranenburg (1974a) used a numerical filter to suppress spurious oscillations. Bergant (1992), Bergant and Simpson (1994b), and Simpson and Bergant (1994a) compared a number of pipe column separation models. They found that within the MOC the staggered grid is preferred above the rectangular grid, which actually comprises two independent staggered grids (chess-board instability). The model with staggered grid and a ψ value of 1.0 (DVCM) gave reasonably accurate results when the number of reaches was restricted (the ratio of maximum cavity size to reach volume should be below 10%).
32
3.4.2
The discrete gas cavity model (DGCM)
Brown (1968) presented the first attempt at describing liquid column separation with the effects of entrained air. Entrained air was assumed to be evenly distributed in concentrated pockets at equal distances along the pipeline. The presence of air decreased the overall pressure wave celerity. A pressure-volume relation was assumed to describe the behavior of the concentrated air pockets, whose locations were assumed to be permanent with no change due to the prevailing direction of flow. The presence of entrained air was neglected above a certain head, where the solution reverted back to normal water hammer computations. The DGCM is a numerical model that has been used for modeling column separation over the last 30 years but not as widely as DVCM. The numerical model utilizes free gas volumes to simulate distributed free gas (Wylie 1992). After Brown (1968), this model was further developed by De Vries (1973), Provoost (1976), Provoost and Wylie (1981) and Wylie (1984). The discrete free gas cavity model is a modification of the discrete vapor cavity model. Figure 8 shows that the DVCM is a limit-case of the DGCM. Provoost (1976) and Wylie (1984) presented a detailed description of a DGCM, in which cavities were concentrated at “grid points” (or computational sections). Pure liquid was assumed to remain in each computational reach. A quantity of free gas was introduced at each computational section. Wylie (1984) and Wylie and Streeter (1993) described the discrete free gas model for simulating vaporous and gaseous cavitation. Gas volumes at each computational section expanded and contracted as the pressure varied and were assumed to behave as an isothermal perfect gas. This model exhibited dispersion of the wave front during rarefaction waves and steepening of the wave front for compressive waves. Distributed cavitation in pipelines may be successfully simulated by using very small quantities of free gas. According to the ideal gas law, the void fraction α varies inversely with the absolute partial pressure of the gas p *g as
α=
M g Rg T
(13)
p*g ∀m
where Mg = mass of free gas, Rg = specific gas constant, T = absolute temperature, and ∀m = unit mixture volume. The amount of gas concentrated at each computational section is determined by coalescing to the section the distributed gas from the adjacent half reaches. The volume of fluid mixture ∀m in each adjacent reach is taken to be constant. Eq. (13) can be written for a reference void fraction α0 at a given reference
33
pressure p*g 0 . As the absolute pressure p *g of the gas changes throughout the transient, the volume of isothermal gas from Eq. (13) and from the reference condition is
∀ g = α ∀m =
p*g 0α 0∀m p
* g
=
C H P − z + H b − pv* / γ
(14)
where C = p*g 0 α0 ∀m / γ, HP = piezometric head at the computational section, z = elevation of the centerline of the pipe, Hb = barometric pressure-head, pv* = absolute vapor pressure at temperature T, and γ = specific weight of liquid. Note that p* = pg* + pv*. A value of the reference void fraction α0 of 10-7 or smaller is taken for simulation of vaporous cavitation. For this order of void fraction, there is only a small change in pressure wave speed due to the presence of the free gas, even at low pressures. Continuity of the discrete gas cavity volume (also the vapor volume) is given by d∀ g dt
(15)
= QP − QP u
where QP and QPu are the flow rates at the downstream and upstream side of the computational section, respectively. Provoost and Wylie (1981) presented various schemes for integrating the gas volume equation (15). The trapezoidal rule yielded high frequency pressure oscillations, whereas a forward integration procedure damped out the oscillations. Wylie (1984) integrated Eq. (15) and introduced a time-direction weighting-factor ψ to control numerical oscillations. The solution written for the staggered grid of the MOC was ∀tg = ∀tg− 2 ∆t + {ψ (QPt − QPt u ) + (1 −ψ ) (QPt − 2 ∆t − QPt −u2 ∆t )} 2∆ t
(16)
in which ∀gt and ∀gt−2∆t are the gas volumes at the current time and at 2∆t earlier. The range of values of ψ was from zero to unity, however, selecting a value greater than 0.5 was recommended to avoid severe oscillations. The forward integration scheme (this is the Euler-backward method for ODE Eq. (15)) used a value of ψ of 1.0 resulting in minimal numerical oscillations (Bergant 1992; Wylie and Streeter 1993; Simpson and Bergant 1994a). Liou (1999, 2000) recommended a value of ψ close to or equal to unity in combination with a sufficient number of computational reaches. In a rigorous treatise of the numerical method, applying Von Neumann analysis to a linearized set of equations, Liou nicely showed that the numerical wave speed converges to the theoretical (physical) one (see Figure 12). He explained why the DCGM exhibits non-linear variable wave-speed features.
34
Figure 12 Wave speed ratios (Liou 2000, Fig. 5). Dots: numerical (DGCM) wave speed / gas-free wave speed. Lines: theoretical mixture wave speed / gas-free wave speed. Wylie and Streeter (1993, pp. 195-196) compared the performance of the DGCM against experimental results (Fig. 13) where only one vapor cavity formed adjacent to the valve with no distributed cavitation. The results clearly show the occurrence of short-duration pressure pulses of different relative magnitudes and widths.
Figure 13 DGCM results versus experimental results (Simpson 1986, Fig. 7.8). Hydraulic grade line (HGL) variation with time. Wylie (1992) and Wylie and Streeter (1993, pp. 202-205) compared results from the DGCM with analytical and experimental data in a low void-fraction system during rapid transient events. Favorable comparisons were obtained although highly non-linear behavior was observed. The wave speed variation with pressure in a system with free gas was demonstrated. Wylie and Streeter (1993, pp. 188-192) tested the DGCM for a bubbly flow case in which air was dispersed in the continuous liquid phase. The experimental results were
35
taken from Akagawa and Fujii (1987). Wylie and Streeter (1993) considered three different void fraction distributions in applying the DGCM. The comparison of numerical and experimental results for two of the three cases was quite favorable despite the results being extremely sensitive to the amount of free gas in the system. In the third case, with a uniform void fraction in one section and a single phase upstream, the DGCM produced oscillations that were not present in the experimental results. Bergant and Simpson (1999a) validated the DGCM (including DVCM and GIVCM (generalized interface vaporous cavitation model) results) against experimental results for the rapid closure of a downstream valve. First the results for two different initial flow velocities (V0 = 0.30 and 1.40 m/s) in an upward sloping pipe were compared. The low velocity case represented a column separation event with a maximum pressure larger than the Joukowsky value (due to a short-duration pressure pulse); the high velocity column separation case generated a maximum pressure lower than the valve-closure pressure. Next the results for two different static heads (H0 = 12.0 and 22.0 m) in the upstream reservoir and an initial flow velocity V0 = 0.30 m/s were considered. Valve closure for the lower static head generated column separation with a wide short-duration pressure pulse. The influence of pipe slope was investigated by comparing experimental runs in a downward and an upward sloping pipe for an identical initial flow velocity V0 = 0.71 m/s and identical static pressure-head at the valve of 20.0 m. The authors found a slightly different timing of the cavity collapse at the valve and a slightly changed intensity of cavitation along the pipeline for the two runs. The differences were caused by the difference in steady head envelope (hydraulic grade line) and direction of action of gravity. In addition, the authors presented a global comparison of DGCM and experimental results for a number of flow regimes in downward and upward sloping pipes (30 cases). A comparison was made for the maximum head at the valve and the duration of maximum cavity volume at the valve. The agreement between the computed and measured results was acceptable. Barbero and Ciaponi (1992) performed a similar global comparison analysis between DGCM simulations and measurements. 3.5 Shallow Water Flow or Separated Flow Models This type of numerical model describes liquid column separation or cavitation regions with shallow water (open channel) theory. The water hammer (liquid) regions are calculated by the method of characteristics. They are separated from the shallow water regions by moving boundaries. Shallow water flow modeling of regions of vapor pressure provided the first real attempt at a more realistic description of transient cavitation. Vapor bubbles were assumed to form, rise quickly and agglomerate to form a single long thin cavity when the pressure reached the vapor pressure (Provoost 1976). Li (1962, 1963) presented a study of the mechanics of pipe flow following a local liquid column separation at an upstream closing valve. A numerical example of the motion and spreading of the vapor-liquid interface was considered when an upstream valve closed instantaneously on an upward sloping pipe of
36
3.9 m length. A topological study of the phase plane was used. Spreading of the interface was described by shallow water theory using two quasi-linear partial differential equations. The study revealed that the spreading of the surface may be neglected in computing the water hammer pressure resulting from the cavity collapse. A small variation in vapor pressure occurred during the existence of the cavity (Li 1962, 1963), which nevertheless may be regarded as constant (Li and Walsh 1964). Rigid liquid column theory was used, which required the assumption that the time of existence of the cavity was long compared with 2L/a. In a Ph.D. thesis at The University of Michigan, Baltzer (1967a) developed a shallow water flow numerical model for column separation at a valve while the water hammer equations were applied in the remaining part of the pipe. The shape, movement, and collapse of a vapor cavity formed at the upstream side of a valve were considered. Once the vapor cavity formed, it usually expanded and propagated in the direction of the flow as a bubble. A detailed description was presented of the sequence of events during the genesis, growth, and collapse of a cavity at a valve. This is illustrated in Figure 14.
Figure 14 Simulated transient pressures at gate valve and concurrent, free-surface profiles at vapor cavity (Baltzer, 1967, Fig. 4). Siemons (1966, 1967) at Delft Hydraulics Laboratory also developed a shallow water flow model of liquid column separation to describe cavitating flow. This model was referred to as a “separated flow” model by Vreugdenhil et al. (1972). The thickness of the cavity was assumed to be small in comparison to the diameter of the pipe. A Lax finite-difference scheme was used to solve the equations. Siemons (1967) stressed that the rise in pressure at the collapse would not be great. Dijkman (1968) and Dijkman and
37
Vreugdenhil (1969) extended Siemons' (1967) “separated flow” model, by considering the case of gas release into a single cavity at a high point in a pipeline. The pressure rise, after compression of the cavity incorporating dissolved gases, was concluded to be less serious than for the vapor-only case. Gas flow equations were solved in conjunction with the shallow water flow equations. The method of characteristics was applied to solve the fourth-order hyperbolic system. The authors concluded that it was uncertain how the collapse pressures may be computed and suggested an approach of attempting to prevent the occurrence of cavitation. Kalkwijk and Kranenburg (1971) noted that Siemons' results did not maintain a mass balance at the boundary of the cavity, and therefore they questioned the validity of the conclusion concerning the generation of high pressures. The transition from the water hammer region to the cavitating region was one of the major problems with the shallow water flow approach. For this reason, Kranenburg (1974a) concluded that the description of cavitating flow and liquid column separation by shallow water flow theory did not seem attractive. Another problem concerned the appearance of gravity waves. Furthermore, the model was concluded to be physically incorrect for vertical pipes. Marsden and Fox (1976) presented a numerical model in which the low-pressure flow region was described by the partial differential equations of open channel flow. The authors used a similar approach to that taken by Baltzer (1967a, 1967b). No evidence of prediction of high pressures was found in their study and the authors concluded that uneconomic overdesign of pipelines resulted if the cavity collapse was computed by normal techniques. Fox and McGarry (1983) presented a variant of the discrete cavity model with a cavity assumed to occupy the upper portion of the pipe and to be spread over a ∆x length. Thermodynamic effects were included, but the authors concluded that their influence was insignificant if the vapor pressure of the liquid was small. 3.6 Two Phase or Distributed Vaporous Cavitation Models This type of numerical model distinguishes between water hammer regions (with pure liquid) and distributed vaporous cavitation regions (with a homogeneous liquid-vapor mixture). Velocity and vapor void fraction are computed in the mixture region. The void fraction is defined as
αv =
∀v ∀m
(17)
where ∀v = volume of vapor and ∀ m = volume of mixture of liquid and vapor.
38
In the early 1970s research related to pipeline cavitation was carried out at Delft Hydraulics Laboratory and at Delft University of Technology in The Netherlands. Kalkwijk and Kranenburg (1971, 1973) presented a theory to describe the occurrence of distributed vaporous cavitation in a horizontal pipeline. They referred to this as the “bubble model”. Their approach ignored free gas content in the liquid and the diffusion of gas towards cavities. An upstream end pump failure was considered. Wylie and Streeter (1978b) presented a similar analytical development of a model for vaporous cavitation in a horizontal pipeline. An example was considered that involved the rupture of a pipeline at the upstream reservoir. Kalkwijk and Kranenburg (1971) presented two approaches to the theory. The first approach was based on the dynamic behavior of nuclei or gas bubbles. However, the method failed at the point where the radii of the bubbles exceeded a critical value. At this size the bubbles became unstable and the characteristics became imaginary. The incorporation of added mass might solve this problem (Geurst 1985). The second approach distinguished between the regions with and without cavitation. Different systems of equations held for the water hammer and the vaporous cavitation region (Kranenburg 1972). The wave celerity in the cavitation region, with respect to the fluid particles, is reduced to zero. Kalkwijk and Kranenburg (1971) assumed that the liquid in a cavitation region had a pressure equal to the vapor pressure. Analytical expressions were developed for the velocity and void fraction for the vaporous zone in a horizontal pipe. When a cavitation region stopped growing, a shock formed at the transition from the water hammer to the vaporous region, which penetrated into the cavitation region. This interface was described using the laws of conservation of mass and momentum, which resulted in “shock equations” analogous to the equations for a moving hydraulic jump. A shock-fitting technique was used (Kranenburg 1972). Use of analytical methods for the treatment of the cavitation region and the explicit shock calculation was implemented to avoid numerical distortions (Vreugdenhil et al. 1972). A number of systematic computations for a simple horizontal pump discharge line showed that the maximum pressure after cavitation did not exceed the steady state operating pressure for the pump. In a Ph.D. thesis at Delft University of Technology, Kranenburg (1974a) presented an extensive work on the effect of free gas on cavitation in pipelines. A detailed description was given of the phenomenon of transient cavitation in pipelines. A simplified one-dimensional mathematical model, referred to as the “simplified bubble flow” model, was presented (Kranenburg 1972, 1974b). Continuity and momentum equations in conservation form were presented for the vaporous regions. The slope of the pipe and the influence of gas release were both considered. Kranenburg (1974a) found that there was considerable difficulty in using the method of characteristics due to the pressure dependence of the wave celerity because of the presence of free gas. He asserted that discontinuities or shocks between the water hammer and vaporous region should be fitted in the continuous solution only for simple cases. To simplify the modeling approach, the bubble flow or vaporous cavitation regime was assumed for the whole pipe, even for the water hammer regions. A modified surface tension term was used to achieve this simplification. As
39
a result, this model did not show explicit transitions between the water hammer and vaporous cavitation regions. Kranenburg (1972) used a Lax-Wendroff two-step scheme despite the occurrence of shock waves. A numerical viscosity was used to suppress the non-linear instability resulting in the spreading of the developing shock wave over a number of “mesh points” as described by Richtmeyer and Morton (1967). In addition, a smoothing operator was introduced to reduce oscillations and instability of computations caused by the pressure dropping to vapor pressure. Liquid column separation was explicitly taken into account at mesh points where it may be expected to occur. Kalkwijk and Kranenburg (1971) also presented experimental results. Dispersion of the negative wave was observed for pressures below atmospheric pressure. This was attributed to the growth of nuclei. Some gas bubbles were observed to remain after the passage of the shock wave that collapsed the vaporous region, and this suggested that gas content played a certain role. Computations did not exhibit the dispersive effect observed for the experimental results. In conclusion, the authors stated “this method gives a reasonable description of the overall behavior of the process”. Fanelli (2000) also addressed methods for solving ordinary and partial differential equations including shock-fitting and shock-capturing procedures. Wylie and Streeter (1978b, 1993) developed a case-specific model involving vaporous cavitation and referred to it as an “analytic model”, while Streeter (1983) referred to a more general model as a “consolidation model”. A distributed vaporous cavitation region (zone) is described by the two-phase flow equations for a homogeneous mixture of liquid and liquid-vapor bubbles (liquid-vapor mixture). A homogeneous mixture of liquid and liquid-vapor bubbles in pressurized pipe flow is assumed to occur when a negative pressure wave traveling into a region of decreasing pressure along the pipe drops the pressure to the liquid-vapor pressure over an extended length of the pipe. Pressure waves do not propagate through an established distributed vaporous cavitation zone.
3.6.1 Two-phase flow equations for vaporous cavitation region
The two equations describing a vaporous cavitation region are the following continuity equation and equation of motion (Bergant and Simpson 1992, Wylie and Streeter 1993) ∂α v ∂α v ∂Vm + Vm − = 0 ∂t ∂x ∂x
(18)
40
f Vm Vm ∂Vm ∂V = 0 + Vm m + g sinθ + ∂t ∂x 2D
(19)
where αv = void fraction of vapor and Vm = liquid-vapor mixture velocity. As the pressure is assumed constant (vapor pressure), only gravitational and friction forces act. The two equations are valid for small void fractions (αv << 1) and up to a temperature of about 330 K (Hatwin et al. 1970). The Darcy-Weisbach friction factor f for liquid flow is assumed in Eq. (19). The friction loss effect due to vaporous bubbles in the liquid-vapor mixture can be ignored for small void fractions (Griffith 1987). Eqs. (18) and (19) can be solved analytically (Streeter 1983). Bergant (1992) and Bergant and Simpson (1999a) presented the solution of the two equations for an upward and downward sloping pipe, and for a horizontal pipe. Introducing the total derivative, the velocity Vm of the liquid-vapor mixture is first calculated from Eq. (19) by analytical integration, then the void fraction αv is estimated by numerical integration of Eq. (18). The solution of Eq. (19) for Vm depends on the pipe slope (upward, downward or horizontal) and the inception (initial) velocity of the liquid-vapor mixture Vmi at time of cavitation inception tin at distance x along the pipeline at which the pressure drops to the liquid vapor pressure. The inception velocity Vmi is calculated from Eq. (9) or Eq. (10) within the MOC numerical grid with the pressure set to the vapor pressure. Noting that Vm is a uniform velocity - independent of x - of one individual distributed cavitation zone, the different results of integration for Vm are as follows (Bergant 1992, Bergant and Simpson 1999a): (1) Sloping pipe with θ Vmi > 0: There are two situations for this case: (i) the pipe slope angle θ is positive and Vmi is positive or (ii) θ is negative and Vmi is negative. (1.1) Before flow reversal: V Vm =Vmt tan tan −1 mi Vmt
fV −sign(θ ) mt (t −tin ) 2D
(20)
in which Vmt = (2gD|sinθ|/f)1/2 = terminal velocity of liquid-vapor mixture in the sloping pipe and sign(θ) = {+1 for θ > 0 or −1 for θ < 0}. (1.2) After flow reversal:
41
Vm = Vmt
e −sign(θ ) f Vmt (t −tr ) / D − 1 e
− sign(θ ) f Vmt (t −tr ) / D
+1
= Vmt tanh{−sign(θ )
f Vmt (t − tr )} 2D
(21)
in which the time of flow reversal tr is:
t r =tin + sign(θ )
V 2D tan −1 mi fVmt Vmt
(22)
(2) Sloping pipe with θ Vmi < 0: There are two situations for this case: (i) θ is positive and Vmi is negative or (ii) θ is negative and Vmi is positive.
Vm =Vmt
Vmi −Vmt +(Vmt +Vmi ) e −sign(θ ) f Vmt (t − t in ) / D
(23)
Vmt −Vmi +(Vmt +Vmi ) e − sign(θ ) f Vmt (t − t in ) / D
(3) Horizontal pipe:
Vm =
2 DVmi 2 D+sign(Vmi ) fVmi (t − tin )
(24)
With Vm given by one of the Eqs (20), (21), (23) or (24), numerical integration of Eq. (18) at time t for the void fraction αv over a time step ∆t assuming a weighting factor ψ in time direction gives:
(α v )k t
= (α v )k
t −∆t
+
{ψ (V ) m
t j +1
− (Vm ) j + (1 − ψ ) (Vm ) j +1 − (Vm ) j t −∆t
t
t −∆t
}
∆t ∆x
(25)
in which j is the number of the upstream node and j+1 the number of the downstream node for the computational reach k of length ∆x (∆x=a∆t).
3.6.2 Shock equations for condensation of vaporous cavitation region
A distributed vaporous cavitation region expands in size as a result of a low-pressure wave propagating into a water-hammer region. Eventually, the distributed vaporous cavitation region stops expanding and the boundary separating the water hammer and vaporous cavitation regions commences to move back into the cavitation region. The progression of the liquid into the liquid-vapor mixture or the collapse of a discrete
42
vapor cavity separating the mixture zone(s) (intermediate cavity) condenses the liquid-vapor mixture back to pure liquid. The liquid is then compressed to a pressure that is unconditionally greater than the liquid vapor pressure. The movement of the interface (shock wave front) separating the one-phase fluid (liquid) and the two-phase fluid (liquid-vapor mixture) is described by shock equations. Isothermal conditions across the interface of infinitesimal width are assumed (Campbell and Pitcher 1958). The shock equations developed for a generalized movement of the interface (movement in either direction along the pipe) are derived from the continuity equation and the equation of motion (Bergant 1992, Bergant and Simpson 1992) are g a s 2 ( H s − H sv ) + α v − (V − Vm ) = 0 a
(26)
g ( H s − H sv ) + (V − Vm )(V − Vm − a s ) = 0
(27)
where as = shock wave speed, Hs = piezometric head on the water-hammer side of the shock wave front, and Hsv = piezometric head on the distributed vaporous cavitation side of the shock wave front. The shock wave equations are coupled with water hammer and two-phase flow equations. Eqs. (26) and (27) form a system of algebraic equations describing the movement of the shock wave front into the liquid-vapor mixture. The shock equations are coupled with Eqs. (9) or (10) depending on the direction of travel of the interface, the kinematic equation for the length of the front movement and the equation of motion for the liquid plug condensed over a part of the reach. Let L be the distance to the shock interface measured from the nearest computational section (through the liquid plug). The kinematic equation for the position of the shock interface as it moves from Lt−∆t to Lt during time step ∆t is: Lt = Lt −∆t + as + Vm ∆ t
(28)
The equation of motion written for the liquid plug of length Lt is:
as as
(H
j
− Hs ) −
f Lt Lt ( Q tj − Q tj−∆t ) = 0 Q j Q tj−∆t − 2 g A∆ t 2 gDA
(29)
in which Hj = piezometric head at the upstream side of the liquid plug at computational section j. Note that Quj = Qj at the upstream end of the liquid plug. The unknowns in the above system are Hj, Hs at the water-hammer side of the interface, Qj, as and Lt. Vm is calculated directly from one of the Eqs. (20) through (24) and αv from Eq. (25). This system of non-linear 43
equations is solved by the Newton-Raphson method (Carnahan et al. 1969). Development of shock equations for the collapse of an intermediate cavity located between two distributed vaporous cavitation regions and for the end boundary conditions (e.g. reservoir, valve) is described in the literature (e.g. Simpson 1986; Bergant 1992; Bergant and Simpson 1992). 3.7 Combined Models / Interface Models This type of model allows for distributed vaporous cavitation in the same way as the previous type, however, the formation of local column separations at any point in the pipeline is taken into account (Kranenburg 1974b, Streeter 1983). Flow regions with different characteristics (that is – water hammer, distributed vaporous cavitation, end cavities and intermediate cavities) are modeled separately, while the region interfaces are tracked (Simpson 1986, Bergant 1992). Kranenburg (1974b) presented consideration of a liquid column separation at a valve, in conjunction with the description of vaporous cavitation regions using his “bubble flow” model. The numerical model was applied to the inclined pipe experiments of Baltzer (1967a, 1967b). Gas release in the vaporous region was concluded to cause damping of the pressure peaks caused by the collapses of liquid column separation. To support this contention a hydrodynamic energy balance was computed for a liquid column separation in a reservoir/horizontal-pipeline/valve system. Contributions to the energy balance included elastic energy of the liquid and pipe wall, elastic energy of the free gas, work done at the reservoir, dissipation caused by shock waves, and dissipation caused by pipe friction. The elastic energy of the free gas was small compared with the dissipation terms. This explained the relatively small influence of gas release at the liquid column separation void. There was only slight damping due to wall friction, noting that the contribution of numerical damping was smaller. The conclusion was drawn that the marked energy loss may be attributed to the dissipation at the shock wave fronts due to heat-transfer and viscosity effects. In summary, Kranenburg (1974b) concluded that the inclusion of gas release had no effect where only cavitating flow occurred, whereas the influence was considerable where liquid column separation occurred in combination with cavitating flow. Gas release in the cavitating flow region adjacent to a liquid column separation diminished the duration of subsequent liquid column separations and thus the maximum pressures upon collapse. Streeter (1983) was the first to develop a combined analysis for modeling local liquid column separations at high points and a number of distributed vaporous cavitation regions, while retaining the shock-fitting approach to explicitly compute the locations of transitions between water hammer and vaporous cavitation regions. Gas release was not considered, thereby removing the problem associated with the variable wave speed due to the presence of free gas. This model was referred to as an “analytical approach” or the
44
“consolidation model”. The model was applicable to pipes at any angle with the horizontal. Many separate distributed vaporous cavitation zones could be modeled, as well as the collapse and reforming of vaporous regions. All water-hammer regions were described by the method of characteristics. The equations developed for the vaporous region were for various combinations of slope and initial (inception) velocity. A computational section became vaporous once a pressure less than the vapor pressure had been computed from the method of characteristics. A time-line interpolation scheme was used to find the first occurrence of vapor at a computational section. The development by Streeter (1983) considered the formation of discrete vapor cavities (column separations) in the pipeline. It was asserted that a vapor cavity may only form if the angle with the horizontal between two adjacent pipe sections decreased in the downstream direction, such as at a high point in the pipeline. If vapor pressure occurred at such a section, a local liquid column separation was assumed to form. For all other pipe slope conditions a vaporous cavitation region was assumed and both the velocity and void fraction in the vaporous region were computed. The model assumed that a vapor cavity could not form at computational sections for which the pipe slope was the same in the upstream and downstream reaches. The same assumption was applied for the case where the angle with the horizontal between two adjacent sections increased in the upstream direction. In making these assumptions, this approach did not account for the possibility of intermediate cavities due to the interaction of the two low-pressure water hammer waves in the pipeline. Wylie and Streeter (1993, pp. 196-207) give a detailed presentation of the so-called interface model for modeling distributed cavitation. The discrete vapor cavity may be located between the following regions (Streeter 1983, Simpson 1986, Bergant 1992, Bergant and Simpson 1992): (1) Two liquid zones, (2) Two distributed vaporous cavitation zones, (3) Liquid and distributed vaporous cavitation zone, (4) End boundary and liquid zone, (5) End boundary and distributed vaporous cavitation zone. Bergant (1992) and Bergant and Simpson (1992) used a standard DVCM algorithm that allows cavities to form at computational sections as a basis for the development of an interface model and referred to it as a generalized interface vaporous cavitation model (GIVCM). Distributed vaporous cavitation zones, shock waves and various types of discrete cavities were important features in modifying the standard DVCM. The model handles a number of pipeline configurations (sloping and horizontal) and various interactions between water hammer, distributed vaporous cavitation, intermediate column separation (along the
45
pipeline), and column separation at boundaries (valve, high point). For example, Figure 15 shows a typical sequence of transient events in a horizontal pipe including growth and collapse of a discrete cavity, propagation of a vaporous cavitation zone along the pipe and two shock-wave fronts. In essence, the GIVCM algorithm maintains the same basic structure as the DVCM and is therefore simpler than previous interface models. A loop for the shock treatment at appropriate computational sections was added to the basic DVCM loop and a module for combined discrete vapor cavity and distributed vaporous cavitation computation was incorporated. Flags to control the correct physical behavior of various phase interactions and to identify possible new interactions support the algorithm.
Cavity collapse t+4∆t Propagation of shock wave front t+3∆t
t+2∆t Propagation of distributed vaporous cavitation zone t+∆t Discrete vapor cavity t j-1
j
j+1
j+2
j+3
j+4
Figure 15 Cavitation and shock formation in a horizontal pipe (adapted from Bergant and Simpson 1992).
Although interface models give reliable results (Bergant 1992, Bergant and Simpson 1999a), they are quite complicated for general use. More accurate treatment of distributed vaporous cavitation zones, shock waves and various types of discrete cavities contribute to improved accuracy of the pipe column separation model. The drawback of this type of model in comparison with discrete cavity models is the complex structure of the algorithm and the longer computation times.
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3.8 Other Models 3.8.1 Jordan algebraic model
Jordan (1965, 1975) developed an analytical method for the treatment of distributed vaporous cavitation zones. He asserted that pressure waves cannot propagate through an established mixture of liquid and vapor (vaporous cavitation zone) and that the Schnyder-Bergeron graphical method could not be used in this region. Jordan developed dynamic equations for the distributed vaporous cavitation zone and equations for the movement of the upstream and downstream liquid columns into the vaporous cavitation region. He calculated the time when the two columns met (condensation of the vaporous cavitation zone) and consequently the re-establishment of the liquid phase. Tarasevich (1975, 1997) developed a similar analytical method. The method was applied to a horizontal pipeline with a valve at the upstream end. 3.8.2 FEM
Howlett (1971) modeled the liquid contained within a pipe system by means of solid beams without bending stiffness in a finite-element (FEM) solution procedure. Watt et al. (1980), Bach and Spangenberg (1990) and Shu (2003a) applied the FEM to the classical water hammer equations. Giesecke (1981) mentioned the discrete cavity model but did not show any results. Axisa and Gibert (1982) and Schwirian (1982, 1984) employed the DVCM within the context of the FEM; gaps were allowed to form between the axial beam elements simulating the liquid. See Figure 16. They compared numerical results obtained with and without cavitation.
Figure 16 DVCM in FEM context (Schwirian 1984, Fig. 4). Dynamic element with gap which allows the formation of a cavity. 3.8.3 Other
Mansour (1996) used the DVCM in combination with a special finite difference scheme for the waterhammer equations. Equations for a simple condenser cooling water system were developed by Fanelli (2000) including unsteady pipe flow equations with variable wave speed and friction losses due to the presence of gas bubbles. Equations for water column separation were presented. In addition, a number
47
of boundary conditions were fully described, including pumps, valves, condensers, siphons, surge tanks and non-return valves. 3.9 A Comparison of Models
De Vries et al. (1971), Kalkwijk et al. (1972) and Vreugdenhil et al. (1972) compared Siemons' “separated flow” model with Kalkwijk and Kranenburg's (1971, 1973) “bubble flow” model. Experimental results were presented for a horizontal 1450 meter test circuit. The construction of some large water supply pipelines in The Netherlands prompted these studies. Allowing the occurrence of pipeline cavitation was considered as an alternative to expensive water-hammer control devices such as surge tanks, air vessels and flywheels. The authors concluded that the results obtained from both computer programs exhibited adequate agreement with the experimental results for the horizontal test circuit. Provoost (1976) also compared the results of the “separated flow” (open channel flow) model and Kranenburg's (1974a) “simplified bubble flow” model for a horizontal pipeline and a pipeline with high points. Provoost (1976) concluded that the “separated flow” model did not reproduce the field measurements for the pipeline system with two high points. The “simplified bubble model” was not suited to describe the local liquid column separations at the high points. This model assumed that the equations for the cavitation region were applied to the entire pipeline. Explicit transitions were not shown between waterhammer regions and vaporous cavitation zones. A filtering procedure was required to suppress a slowly developing instability in the cavitation regions. As a result, the discrete free gas model was developed by De Vries (1973) and Provoost (1976) in order to deal with local liquid column separations at high points. In his Ph.D. thesis, Simpson (1986) compared predictions of both his interface model and the discrete cavity model with experimental data obtained in a 36 m long upward sloping pipe of 20 mm diameter. Bergant (1992) and Bergant and Simpson (1999a) compared numerical results from discrete vapor (DVCM), discrete gas (DGCM) and generalized interface vaporous cavitation models (GIVCM) with results of measurements performed in a 37.2 m long sloping pipeline of 22 mm diameter. The principle source of discrepancies between the computed and measured column separation results was found to originate from the method of physical description of vaporous cavitation zones and resulting phenomena along the pipeline. Dudlik et al. (2000) compared the DVCM with a three-phase model that allowed for the calculation of sudden changes of gas content in the liquid. Shu (2003b) compared the DVCM with a two-phase model and with experimental data from Sanada et al. (1990).
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3.10
Modeling of Friction
The friction term in unsteady pipe flow differs from the friction term in steady pipe flow. Experimental validation of quasi-steady friction models for rapid transients without column separation has shown significant discrepancies in attenuation and timing of pressure histories. Computational results have been compared with the results of measurements by Holmboe and Rouleau (1967), Vardy (1980), Brunone and Greco (1990), Golia (1990), Bergant and Simpson (1994a), Brunone et al. (1995), Bughazem and Anderson (1996), Bergant et al. (2001) and Shu (2003b). The quasi-steady friction model may be used for transients where the wall shear stress is in phase with the flow velocity (cross-sectional average). Traditionally, the quasi-steady friction term has been incorporated in most column separation models. This term may contribute to inaccuracies of numerical model results. The influence of unsteady friction effects during column separation has not been studied in any depth. Researchers have attempted to incorporate a number of unsteady friction models for liquid flow into standard discrete cavity models that are used in most engineering transient simulation software packages (Safwat et al. 1986; Dudlik 1999). The numerically stable waterhammer compatibility equations, written in a finite-difference form for a computational section with index i, are: − along the C+ characteristic line (∆x/∆t = a):
H tj − H tj −−1∆t +
f ∆x a {(Qu ) tj − Q tj −−1∆t } + (Qu ) tj Q tj −−1∆t = 0 gA 2 gDA2
(30)
− along the C− characteristic line (∆x/∆t = −a):
H tj − H tj −+1∆t −
a f ∆x {Q tj − (Qu )tj−+∆1t } − Q tj (Qu )tj−+∆1t = 0 gA 2 gDA2
(31)
The unsteady friction factor f used in Eqs (30) and (31) can be expressed as the sum of a quasi-steady part fq and an unsteady part fu, i.e. f = fq + fu (Zielke 1968; Vardy 1980; Vardy and Brown 2000; Bergant et al. 2001). Numerous unsteady friction models have been proposed to date including one-(1D) and two-(2D) dimensional models (Stecki and Davis 1986; Brereton and Mankbadi 1995; Gündogdu and Çarpinlioglu 1999). The 1D models approximate the actual cross-sectional velocity profile and the corresponding viscous losses in different ways. Shuy and Apelt (1983) performed numerical analysis with five different friction models (steady, quasi-steady and three unsteady friction models - those of Carstens and Roller (1959), Trikha (1975) and Hino et al. (1977) -
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that had been incorporated into a standard DVCM. The authors studied 'slow' transients in two long pipelines (2.3 km and 9 km). They found small differences in the results of the five models for the case of pure water hammer. Large discrepancies between the five different results occurred for the column-separation case. Column separation leads to very fast variations in pressure and flow velocity when voids collapse. Brunone and Greco (1990) and Golia and Greco (1990) used a DVCM in combination with Golia's (1990) unsteady friction model. Numerical results were compared with results of measurements of 'rapid' water hammer and of column separation. Significant discrepancies between experiment and theory were found for all runs when using a quasi-steady friction term. Golia's model results showed an improved agreement between the computed and measured results. The agreement was better for the water hammer case than for the column separation case. Brunone et al. (1991) applied their unsteady friction model in the DVCM. The results for the column separation case showed similar behavior as the results with Golia's model. Bergant and Simpson (1994a) investigated the performance of the quasi-steady and the three distinct types of unsteady friction models - those of Zielke (1968), Hino et al. (1977) and Brunone et al. (1991). The friction models had been incorporated into a standard DVCM using the staggered grid of the method of characteristics. Results of calculations were compared with experimental results for a fast valve-closure in a reservoir-pipeline-valve system. The Zielke and the Brunone (et al.) unsteady friction models gave the best fit for the case of no column separation (water hammer). The effectiveness of the numerical models differed for the case of column separation because the DVCM is inconsistent due to multi-cavity collapse (Simpson and Bergant 1994a). Bughazem and Anderson (2000) extended the earlier study by Anderson and Arfaie (1991) to the modified DVCM that simulates apparently variable wave speed. Their study showed that the effectiveness of the Brunone unsteady friction model did not appear to be influenced by the choice of the column separation model. 3.11
State of the Art - The Recommended Models
The discrete vapor cavity model (DVCM) gives acceptable results when clearly defined isolated cavity positions occur rather than distributed cavitation. The DVCM and its variations like DGCM involve a relatively simple numerical algorithm in comparison to the interface models. The latest developments that were aimed at models physically better than DVCM and DGCM are due to Simpson (1986) and Bergant (1992). Simpson's work (Simpson 1986; Simpson and Wylie 1987, 1989, 1991; Bergant and Simpson 1992) is based on that of Streeter (1983). However, Simpson’s (1986) interface model allowed column separations to form at any point in a pipe system whenever two low-pressure waves meet. Furthermore, the GIVCM (Bergant 1992; Bergant and Simpson 1999a) enables direct tracking of actual column separation phenomena (e.g. discrete cavities, vaporous cavitation zones) and consequently it gives better insight into the transient event. The discrepancies between measured data and DVCM, DGCM and GIVCM predictions found by temporal and global comparisons (Bergant and Simpson 1999a) may be attributed to approximate modeling of column separation along the pipeline (distributed vaporous cavitation region, actual number and position of
50
intermediate cavities) resulting in slightly different timing of cavity collapse and different superposition of waves. In addition, discrepancies may also originate from discretization in the numerical models, the unsteady friction term being approximated as a quasi-steady friction term, and uncertainties and stochastic behavior in the measurement (Simpson and Bergant 1996). At the present stage, the GIVCM is used as a research tool whereas the DVCM and DGCM models are used in most commercial software packages for water-hammer analysis.
3.11.1 Discrete cavity models: limitations
When the absolute pressure reaches the vapor pressure, cavities or bubbles will develop in the liquid. In the DVCM these cavities are concentrated, or lumped, at the grid points. Between the grid points pure liquid is assumed for which the basic water hammer equations remain valid. This means that the pressure wave speed a is maintained (and convective terms neglected) between grid points in distributed cavitation regions. However, in bubble flow the pressure wave speed is very low and pressure-dependent. These matters are implicit in the model (Liou 1999, 2000). Pressure waves actually do not propagate through an established distributed cavitation region, since this is at an assumed constant vapor pressure. The annihilation of a distributed cavitation region by a pressure wave causes a delay in propagation, which must be regarded as a reduction of the wave speed. In the DVCM the cavities, concentrated at grid points, do not move. This is consistent with the acoustic approximation: since the overall time scale is acoustic (water hammer), the displacements of vapor bubbles are small. Vreugdenhil (1964) took into account, within the DVCM, the motion of liquid-vapor boundaries. The maximum length, L vc = ∀v / A , of a cavity must be small compared to the spatial grid size. Simpson and Bergant (1994a) recommended
L vc < 0.1 ∆x
(32)
where ∀v = the volume of the vapor cavity, A = area of the pipe and ∆x = reach length used in the numerical simulation. For distributed cavitation regions, condition (32) is mostly fulfilled. If not, the DVCM model is not valid and the application of models for two-phase plug flow, slug flow or open-channel flow should be considered. For column separations, condition (32) may sometimes be violated, which is acceptable in the opinion of the authors, since a column separation is a local phenomenon, and only a few grid points are concerned. However,
51
care should be taken when L vc / ∆ x becomes larger than 1. In that case liquid-vapor boundaries moving from grid point to grid point should be explicitly modeled. The mass of vapor is neglected, just as the influence of radial pipe displacements on the cavity volume. From a physical "microscopic" point of view the "macroscopic" DVCM is not correct; a two-phase flow (Wallis 1969) approach would be better. However, during cavitation, p* = p v*
(33)
is physically a strong condition, if there is no free gas or gas release involved. Furthermore, the continuity equation, that is the mass balance, is satisfied throughout. As stated, the discrete cavity model is a relatively simple model, which is able to cover the essential phenomena in transient cavitation. It fits in with the standard MOC approach, so that it can be used in general waterhammer computer-codes. Its main deficiency is in the numerical oscillations and unrealistic spikes appearing in the calculated pressure histories, when regions of distributed cavitation occur (De Vries 1972, Bergant and Simpson 1999a). One way of partly suppressing the oscillations and spikes (see Section 3.4.1) is by assuming small amounts of initial free gas in the grid points (De Vries 1973; Provoost 1976; Wylie 1984; Zielke and Perko 1985; Simpson 1986; Barbero and Ciaponi 1991; Bergant 1992; Simpson and Bergant 1994a). Condition (Eq. 33) is then replaced by ( p* − p v* ) ∀ g = constant
(34)
where the free gas is assumed to behave isothermally. The recommended free gas void fractions to be used are of the order of 10−7. The numerical integration of equation (Eq. 15) may also affect the amount of oscillations and spikes (Provoost and Wylie 1981; Simpson and Wylie 1985; Simpson and Bergant 1994a; Liou 1999, 2000). The application of a numerical filter may be considered (Vliegenthart 1970; Kranenburg 1974a). Bergant (1992), and Simpson and Bergant (1994a), who performed a systematic study of the numerical oscillations and non-physical spikes, came to the conclusion that the discrete cavity model does not converge and, consequently, the number of grid points should be limited. Liou (1999, 2000), however, showed that the DGCM converges with respect to the wave speed (Fig. 12). This is why the DGCM can capture the steepening of a positive pressure wave and the spreading of a negative pressure wave. The opinion of the authors is the following. Numerical oscillations and spikes are due to multi-cavity collapse during the annihilation of a cavitation region. In fact, discrete cavity models assume small column separations at every grid point in such a region. Upon collapse of a single column separation a pressure rise or spike occurs, which, once generated, does not disappear. The spike travels in a cavitation-free region to and from the 52
boundaries and when two of these spikes meet a higher spike is the result. However, the one-dimensional longwavelength water-hammer theory is not valid for these short-duration pressure spikes. (Note that the duration of the spikes decreases when the computational grid is refined.) Physical dispersion should be introduced in the model, so that spikes travel with attenuation and more realistic results are obtained. On the other hand, highfrequency oscillations are to a certain extent observed in cavitation measurements: they are unstable and physically not repeatable (Fan and Tijsseling 1992; Simpson and Bergant 1996). This instability, reflected in the unstable numerical oscillations of the discrete cavity models, is part of the cavitation phenomenon. It is acceptable to smooth the grid-dependent highest frequencies by means of a numerical filter. The overall conclusion is that discrete cavity models are adequate, but improvements concerning the numerical oscillations are welcome, because engineers tend to take the highest pressure, which might be an unrealistic peak, as a measure for design and operation. Unsteady friction models help to predict more accurately the time intervals between successive column separations.
4
EXPERIMENTS IN THE LABORATORY AND MEASUREMENTS IN THE FIELD
In all experiments described in this section: unless otherwise stated, water hammer is initiated by the rapid closure of a valve. 4.1 Photographs of Cavity Formation
Escande and Nougaro (1953) reported an early flow visualization study. They conducted column separation experiments in a laboratory apparatus comprised of a 25 m long horizontal pipeline of 200 mm internal diameter. A transparent section was positioned next to the valve for flow visualization using a high-speed camera with 1700 images per second. A large vapor cavity was observed following the closure of the valve. Bunt (1953), Smirnov (1954) and Blind (1956) made similar observations; they presented photographs exhibiting the shape of the cavity at the valve during its growth and collapse. The vapor-liquid interface sloped gently over quite a long distance. Duc (1959) investigated liquid column separation, for three different piping configurations at a high point, due to the shut down of pumps in a 1 km long discharge line at a field installation. A clear piece of Plexiglas pipe at the high point allowed a sequence of photographs to be taken, which exhibited the changes in cavity during a local liquid column separation. These tests contradicted the generally accepted supposition of a complete dynamic separation of the liquid column at an elevated point in the line. Liquid remained in the pipe at the separation point until return-flow filled the void.
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O'Neill (1959) presented some photographs from visual studies using a high-speed movie camera. He concluded that intermediate cavities did not form over the entire pipe cross section, as ideally assumed in the analytical analysis. Most cavities appeared in the upper portion of the cross section only. He concluded that the existence of any shape of intermediate cavity acted to form a control point in the pipeline, but the shape might have had an effect on the subsequent pressure surge generated by the cavity collapse. Li and Walsh (1964), Baltzer (1967b) and Safwat (1972a) presented photographs of a discrete cavity at the downstream side of a closing valve upstream in a simple pipeline system. The appearance of tiny bubbles was observed across the whole cross section, which extended along a large portion of the pipe. Pressure peaks due to cavity collapse were also presented. Tanahashi and Kasahara (1970) studied the formation and collapse of a discrete cavity at the high point of a pumping system. Nonoshita et al. (1991, 1992, 1999) presented photographs of liquid column separation in a laboratory draft tube of a water turbine following wicket gates closure. They studied the effect of the draft tube inlet swirl on column separation events. The swirl flow generated gas release and subsequent attenuation of maximum pressure following cavity collapse. Dudlik et al. (1997, 1999) and Dudlik (1999) presented photographs of column separation in a 230 m long test rig with complex pipe geometry. Swaffield (1969-1970) and Kojima et al. (1984) employed photography to visualize column separation in liquids other than water (kerosene, mineral oil). Differences in the column separation mechanism for different liquids were not observed. The development of new flow visualization technologies in the 1990s included high-speed video and electrical capacitance tomography techniques. Bergant (1992) and Bergant and Simpson (1996) presented photographs of column separation in their sloping pipeline experimental apparatus using high-speed videoequipment. The system comprised a polycarbonate flow-visualization pipe section, high-intensity illuminator and high-speed video Kodak Ektapro 1000. A discrete vapor cavity and a vaporous cavitation zone were observed following rapid valve closure. Adam et al. (1998) used an electrical capacitance tomography technique to observe cross-sectional images from changes in fluid capacitance in a 31.5 m long pipeline of 41.6 mm diameter. A similar approach was used by Dudlik et al. (1997, 1999) and Dudlik (1999) and referred to as 'wire-mesh' visualization technique. 4.2 Laboratory Experiments with Liquid Column Separation
At the end of 19th century Joukowsky (1900) performed water hammer and column separation experiments in Aleksejew's water supply system in Moscow, Russia. He connected three pipelines ((L, D) = ((320, 51); (320, 102); (325, 152)) (m, mm)), each with a downstream valve, to the main supply pipeline of diameter 610 mm. The pressure records were taken at the valve and along the pipeline. Joukowsky was the first to
54
qualitatively explain water column separation phenomena in pipelines (see Section 2.2). In his paper, he noted that it was Carpenter (1894) who was the first one to record sub-atmospheric pressures. Langevin (1928) was probably the first person using piezo-electric pressure transducers for the measurement of column separation. His record shown in Fig. 17 is typical for all later measurements.
Figure 17 Typical pressure history for repeated column separation and collapse (Bergeron 1950, Fig. 56). Binnie and Thackrah (1951) performed an investigation that used an experimental apparatus consisting of a pipeline with an automatic air-inlet valve. When the air-valve was removed, a series of violent impacts took place following a local liquid column separation and the collapse of a vapor cavity. Maximum recorded pressures were actually somewhat greater than the theoretical estimates. Pressure traces indicated the existence of a short-duration pressure pulse due to cavity collapse, which exceeded the “main shock pressure”. The authors attributed the higher pressures to the existence of additional pressure waves caused by reflections from bends, sockets and other discontinuities in the pipeline. Their analysis was based on rigid liquid column theory and on elastic column theory. Lupton (1953), in discussing Binnie and Thackrah's (1951) paper, did not agree with the use of air valves because of the unpredictably high pressures that may result from the presence of air in a pipeline. Bunt (1953) presented findings of a laboratory investigation that considered the possibility of occurrence of liquid column separation during a water hammer event. In addition to local liquid column separations at the valve and at high points, the possibility of the formation of distributed vaporous cavitation region was recognized.
Carstens and Hagler (1964, 1966) considered water hammer in a pipeline transporting phosphate-ore slurry. Results were presented from both a model study and an analytical analysis that described the liquid column separation phenomenon. Jordan (1965) investigated column separation and distributed vaporous cavitation in a laboratory apparatus comprising a 202 m long upward sloping pipeline of 52 mm diameter. The first 25.3 m of the pipe had a slope of 28% and the rest a slope of 1.9%. Baltzer (1967a, 1967b) tested in a coiled copper-pipe laboratory apparatus and he found that his experimentally measured pressure rises were appreciably smaller than the simulated ones. In addition, the recurrence intervals between successive
55
experimental pressure rises were of appreciably shorter duration than the predicted intervals. Baltzer concluded that both of these differences pointed to “higher-than-anticipated” energy dissipation and believed it to be the result of highly turbulent, meta-stable, two-phase flow that occurred ahead of the main column separation. Tanahashi and Kasahara (1970) presented a comparison of experimental and analytical results of water hammer with a local liquid column separation. Liquid column separation occurred at the valve and at the pipe mid-point. Photographs were also presented of the growth of a cavity at a high point. Safwat (1972a) performed measurements in a 46 m long horizontal Plexiglas pipeline of 90 mm diameter. A quantitative analysis of pressure records and photographs of column separation was presented. In addition, Safwat (1972b, 1972c) and Safwat and de Kluyver (1972) presented results of a comprehensive experimental study of pressure surges in a laboratory condenser apparatus. Vreugdenhil et al. (1972) conducted tests in a 1450 m long horizontal pipe of internal diameter of 100 mm. Cavitation was generated by the sudden decrease of pressure at the upstream end of the pipeline. Piga and Sambiago (1974) tested a laboratory cooling water system with three possible configurations (length 25 to 30 m, diameter 100 mm). Thorley and Chohan’s (1976) experimental test facility comprised a horizontal pipe set-up (length 16.3 m, diameter 38 mm) for the study of compressive and rarefaction waves. Provoost (1976) added two high points to the original apparatus of Vreugdenhil et al. (1972) and performed the tests. Krivchenko et al. (1975) presented results of column separation measurements in a draft tube of the Kaplan turbine test rig at MISI, Moscow, Russia. A large cavity formed after rapid closure of the wicket gates (closing time of 0.085 s). Column separation first occurred in the space between the guide vanes and the runner, the cavity then grew into the draft tube inlet cone. A large pressure pulse occurred after cavity collapse. The authors provided measurements of the pressures under the turbine head cover and the draft tube inlet, as well as measurements of the axial hydraulic force acting on the turbine runner. Nonoshita et al. (1991, 1992, 1999) performed similar experimental tests. Katz and Chai (1978) carried out experiments in 0.3 m long tubes of 5 or 6 mm diameter with 2 ms valve closures. They showed one diagram of column separation duration as a function of initial velocity ranging from 1 to 20 m/s. This is the only paper describing a beneficial use of column separation, namely as feedback mechanism in liquid oscillators that can be used for industrial cleaning. Their theory, based on rigid column theory, included the effect of gas release. Kot and Youngdahl (1978a, 1978b) gave a clear explanation of a discrete cavity model and used experiments in a 9.15 m long closed tube for validation. Aga et al. (1980) applied the DVCM to oil flow in a 250 m long, 90 mm diameter, test rig. Van De Sande and Belde (1981) tested a U-tube laboratory apparatus (length of approximately 9 m and diameter of 45 mm). They measured pressure peaks higher than the Joukowsky pressure and contributed it to the delay in rheological response of both vapor and liquid. They used discrete cavities in their simulations. Gottlieb et al. (1981) constructed four different configurations of a steel and a plastic pipeline.
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Extremely high pressure peaks were recorded immediately upon collapse of the vapor cavities. The pressure dropped to about 40% of the pressure peak level and maintained this level for 2L/a seconds. The authors concluded the presence of peaks resembled the pressure peaks associated with the implosion of gas bubbles in pumps. Graze and Horlacher (1983) built two horizontal test rigs (120 m and 86 m, and diameters of 82 mm and 200 mm, respectively). The authors stressed the importance of using adequate pressure transducers and recording equipment to avoid unrealistic pressure spikes, see Section 4.5 herein. Fox and McGarry (1983) developed a test rig for the study of pressure transients in pipelines carrying volatile liquids (18 m length, 55 mm diameter). Martin (1983) tested in a coiled copper tube (102 m length, 13.4 mm diameter). He presented experimental results with limited, moderate and severe cavitation, the severity based on the ratio between the Joukowsky pressure and the tank pressure, see also Section 2.6. Borga and De Almeida (1985) tested in a horizontal pipeline of 105 m length. They concluded that the type of pressure transducer should be a sophisticated low-inertia instrument. In addition, they studied the influence of an in-line non-return valve on reducing the pressure pulses. Simpson (1986) studied short-duration pressure pulses resulting from cavity collapse. An experimental pipeline apparatus was designed and constructed in the G.G. Brown Hydraulics Laboratory at the University of Michigan (USA), in which measurements were done for eight levels of cavitation severity. Experimental results obtained for three levels of cavitation severity were compared with predictions by the DVCM in Fig. 18. The test rig, shown in Fig. 19, consisted of a 36 m long copper pipeline with an inner diameter of 19 mm and a wall thickness of 1.6 mm. The pipeline connected two reservoirs and was upward sloping. The difference in elevation between the downstream end and the upstream end is 1 m, so that the elevation angle θ = – 0.028 radians. The pipeline was rigidly supported every 2.5 m by brackets affixed to a wall. The elbow in the pipeline is also rigidly fixed, by two brackets, so that fluid-structure interaction effects were insignificant. Ordinary tap water was used in the experiments.
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Figure 18 DVCM results against experimental results of Simpson (1986). Gauge pressure-heads at valve for four different steady-state situations (Tijsseling 1993, Fig. 6.1).
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Figure 19 Schematic representation of experimental apparatus (Simpson 1986, Fig. 5.1). Paredes et al. (1987) constructed a 1460 m long and 104 mm diameter horizontal iron pipe apparatus. Carmona et al. (1987) conducted extensive measurements in a laboratory set-up resembling that of Provoost (1976) and they showed the corresponding numerical results. Golia and Greco (1990) found excellent agreement between computations and experimental data provided by Martin (1983). Barbero and Ciaponi (1991) reported on 23 experiments performed in a nearly 500 m long, 110 mm diameter, test circuit. In their calculations they examined the influence of initial free gas and gas release. Anderson et al. (1991) and Anderson and Arfaie (1991) discussed several aspects of the DVCM and showed results of laboratory measurements for three levels of cavitation severity. The laboratory apparatus of Shinada and Kojima (1987, 1989, 1995) was a small-scale physical model of a hydraulic press. The oil-filled test pipe was 5 m long with a diameter of 19 mm. The test results were compared with the results of a single-vapor-cavity model that included laminar unsteady friction and a dynamic equation of motion for the valve.
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Tijsseling and Fan (1991a, 1991b, 1992) carried out measurements in a 4.5 m long stainless steel pipe with two closed ends. The pipe (outer diameter 59.9 mm, wall thickness 3.9 mm) was suspended by thin steel wires and filled with highly pressurized water. Transients were generated through the impact of a solid steel rod at one end of the pipe. The apparatus is very suitable to perform cavitation tests. By taking the initial pressure of the water low enough, the rod impact will cause transient cavitation. Two complications encountered in the conventional reservoir-pipe-valve system are absent: the initial steady state pressure gradient and the nonlinear valve closure. The water is stored under pressure in a closed container so that the amount of free gas is negligible. Due to the time scale of the experiment (in milliseconds): there is no release of dissolved gas and friction effects are unimportant. The way of generating transients leads to very steep wave fronts (pressure rises in microseconds) and the vapor-liquid interfaces at column separations are believed to be nearly perpendicular to the pipe axis. The experiment isolates vaporous cavitation (in combination with fluidstructure interaction). The severity of cavitation was regulated by the initial pressure of the water and the initial velocity of the rod. The experimental results were used to validate the DVCM. From 1989 to 1992, Bergant and Simpson performed a comprehensive experimental test programme for the investigation of column separation events at the University of Adelaide in Australia (Bergant 1992; Bergant and Simpson 1995; Simpson and Bergant 1996). The adjustable experimental apparatus comprised a straight 37.2 m long sloping copper pipeline of 22 mm internal diameter and 1.6 mm wall thickness connecting two computer-controlled pressurized tanks. The pipe slope was constant at about 5.4%. The excitation valve could be located at either end of the pipeline adjacent to either pressurized tank or at the midpoint of the pipeline. The valve is closed by a torsional spring actuator (valve closure times from 5 to 10 ms) and equipped with an optical valve position recording system. A high-pressure flow visualization section enabled a high-speed video to film the growth of transient cavitation. Five pressure transducers were located along the pipeline. The influence of the following quantities on the magnitude, shape and timing of column separation induced pressure pulses was investigated in the experimental programme: (1) Initial flow velocity in pipeline, (2) Static pressure-head in each tank, (3) Pipe slope (upwards, downwards), (4) Position of the fast-closing valve in the pipeline system (downstream end, midpoint, upstream end), (5) Valve closure time. Repeat tests and uncertainty analyses were performed (Bergant and Simpson 1995; Simpson and Bergant 1996). The results and documentation of all 116 measurements are available on CD-ROM and can be obtained through the School of Civil and Environmental Engineering at the University of Adelaide.
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It is difficult to directly measure flow rates in transient flows. Washio et al. (1996) developed a procedure to accurately deduce flow rates in transient laminar flow from two closely located pressure transducers. The procedure is based on frequency-dependent friction theory (Zielke 1968) and relies on a specially constructed differential amplifier. The method was shown to work very well for oil column-separation in the test rig of Washio et al. (1994). The typical changes of flow rates during column separation and rejoining were successfully measured for the first time. Mitosek (1997, 1998, 2000) took cavitation measurements in plastic pipes. He recorded extremely high pressure peaks that might be attributed to the strain-gauge type pressure transducers used (see Section 4.5). Greenshields and Leevers (2000) studied the brittle fracture behavior of plastic pipes. Their laboratory apparatus consisted of a closed vertically falling pipe where column separation occurred at the bottom end. Surge and cavity collapse were able to fracture pipes with a defect. The IAHR working group (Fanelli 2000) is holding experimental records of the laboratory tests by Paredes et al. (1987) performed in a 1460 m long, 104 mm diameter horizontal iron pipeline and totaling 13 experiments with different initial flow velocities (0.35 to 0.82 m/s) and static heads (16 to 80 m). Lai et al. (2000) carried out tests with initial voids of different air content in a pipeline with one vertical branch. Deaerated water was the test liquid. The test results were used to validate their DGCM code, which had a polytropic gas law, but no gas release mechanism. Tabei et al. (2003) added small amounts of the noble gases xenon and argon to initially degassed water to provoke light emission at bubble collapse. In this way, they were able to accurately determine the speed of a shock wave entering a region of bubbly flow. Their theoretical study showed that local temperatures up to 7000 K may occur when small bubbles collapse. 4.3 Gas Release
Keller and Zielke (1976) measured free gas variations subsequent to a rapid drop in pressure in a 32 m long plastic pipe with a diameter of 125 mm that was connected to a cavitation tunnel. Wiggert and Sundquist (1979) conducted experiments using a 129 m and a 295 m long coiled copper-tube apparatus with a diameter of 25 mm. They investigated gas release during transients at different initial gas concentrations. The effects of gas release, cavitation nuclei and turbulence were studied. Martin (1981) used a Plexiglas pipe apparatus (length 32 m, diameter 26 mm) where the water was saturated with injected air. Kazama (1983) performed experiments with water-methanol mixtures. A test section of 2 m length contained an initial cavity of vapor and foam at its vertical closed end. Sudden valve opening created a
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pressure wave collapsing the cavity. The pressure at one location and the volume of remaining air were measured. Shinada (1994) studied experimentally and theoretically column separation with gas release. In a 2.5 m long, 19 mm diameter pipe he tested with saturated and deaerated oil. He measured air content, surface tension and diffusion rate. On the basis of experimental results, the proposed bubble-diffusion model allowed for gas release only at the column separation. Gas release has a significant effect on column separation in saturated oil. 4.4 Fluid Structure Interaction (FSI)
The repeated collapse of column separations, and the almost instantaneous pressure rises associated with them (see Figure 17), form a severe load for pipelines and their supporting structures. Structural vibration is likely to occur. Fluid-induced structural motion, structure-induced fluid motion and the underlying coupling mechanisms are commonly referred to as FSI (fluid-structure interaction). Most of the researchers mentioned in this review paper prevented unwanted FSI effects by rigidly anchoring their pipes. Fan and Tijsseling (1992), however, focused on the simultaneous occurrence of cavitation and FSI. They performed experiments in a closed pipe, the vibrating ends of which interacted with transient column separations. They observed distributed vaporous cavitation caused by a stress wave in the pipe wall. Tijsseling et al. (1996) investigated experimentally and theoretically column separation in a freely moving one-elbow pipe system. More information on combined cavitation/FSI models and on FSI in general can be found in review papers by Tijsseling (1996) and Wiggert and Tijsseling (2001). 4.5 Problems with Pressure Transducers
Care must be taken with the selection (and way of mounting) of pressure transducers to be used in cavitation measurements. Because the collapses of column separations are impact loads for the pipe wall, they must be compensated for accelerations. The pressure transducer's natural frequency must not be too low, thereby noting that exploding and imploding cavitation bubbles may lead to pressure signals with a frequency spectrum up to 1 MHz (Oldenziel and Teijema 1976, p. 14). Graze and Horlacher (1983) and Simpson and Bergant (1991, 1994b) reported unrealistic pressure spikes and oscillations for inductive and strain-gauge type pressure transducers and attributed these to the fact that the natural frequency of the transducer was too low. Sayir and Hausler (1991), who performed cavitation experiments in a 20 m long, 110 mm diameter, closed tube of transparent PVC, had also problems with a too low natural frequency of piezoelectric transducers. Le et al. (1989, p. 3) developed their own special transducers with a natural frequency of 1.7 MHz to overcome this problem. Arndt et al. (1995) described how to make your own pressure transducers. Mitosek (1997, 1998, 2000) found unrealistically high-pressure peaks with straingauge type pressure transducers. Greenshields and Leevers (2000) suggested that the cause may be the
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frequencies in the measured signal in relation to the natural frequency of the piezoelectric pressure transducer. Pressure transducers may be easily damaged during cavitation tests, because nearby explosions and implosions of small bubbles are a too severe load for them (Chen and Israelachvili 1991; Broos 1993, p. 13). 4.6 Field Measurements
Apelt (1956) measured pressures in the field for water hammer in pump discharge lines that undergo liquid column separation. His investigations verified that elastic column theory may be applied with confidence, but that it could not account for the phenomenon of liquid column separation. Richards (1956) also presented some field test data for pumping plants. He contended that it was virtually impossible to analytically analyze such systems. O'Brien (1956) took issue with this point and cited some extensive calculations that involved six or seven separate liquid column separations. Whiteman and Pearsall (1965) conducted field tests at a pumping station on reflux-valve characteristics and pressure rises after pump shutdown. Heavy flywheels were used to lengthen the run-down time and thus prevented liquid column separation. Duc (1959) measured liquid column separation in a field installation. Duc (1965) presented further field test results at the Third Symposium on Water Hammer (Proceedings 1965). Design of protective measures was emphasized to prevent any pressure dropping to vapor pressure. In addition to recording pressure peaks due to rejoining of liquid columns, the phenomena were observed visually. Results exhibited some narrow highpressure peaks of short duration. These may have resulted from the type of pressure recording device being used (see discussion in Section 4.5). Brown (1968) presented field measurements of transients in two pump discharge lines with distinct “knees” in their profiles. A mathematical model was also presented in which a local liquid column separation was assumed to occur at the “knee”. The pressures measured in the field were greater than those predicted by the graphical method in the design stage. This was attributed to the presence of air and gases entrained in solution in the liquid. Brown noted that the presence of air may result in large pressure surges and higher reverse speeds of pumps due to the prolonging of liquid column separation. De Vries (1975a, 1975b) carried out measurements in a thermal power plant cooling water system (pipeline at the upstream end of the condenser: length of 76 m and diameter of 1.8 m; pipeline at the downstream end of the condenser: length of 182 m and diameter of 1.8 m). Provoost (1976) conducted an investigation of a 27.9 km long pipeline with a diameter of 1.8 m. The pipeline was carefully de-aerated before each test. Six cases of pump shutdown were presented. Sharp (1977) presented field measurements in a 12.3 km long, 250
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mm diameter, pipeline. In his calculation he allowed cavities to form at not more than six locations. Nevertheless he obtained reasonable results. Siccardi (1979) presented a comprehensive summary of measurements and computations of water hammer and column separation in a cooling water system at La Casella thermal power plant (pipeline at the upstream end of the condenser: length of 440 m and diameter of 2 m; pipeline at the downstream end of the condenser: length of 234 m and diameter of 1.6 mm). The IAHR working group (Fanelli 2000) is holding the experimental records in the La Casella industrial plant through Daco and Meregalli (1981) who provided 25 experimental runs in which pump start-up and pump power-failure tests at different operating conditions were recorded. De Almeida and Hipolito (1981) performed measurements at a 70 MW thermal power-plant cooling-water system featuring a non-return butterfly valve. Water column separation occurred upstream from the condenser in a series of pump start-up and trip-off tests. Jolas (1981) investigated cavitation effects during pump start-up in a 900 MW Saint-Laurent nuclear (PWR) power plant cooling water system. Enever (1983) performed water hammer and column separation measurements in cooling water systems at the Fawley and the Grain thermal power station. Yow et al. (1985) measured the pressures at two locations after pump restart with a vapor gap in the water cooling system of a nuclear power station. Wang and Locher (1991) found surprisingly good agreement between simulations and field data obtained in a 47 km long cross-country pipeline with a diameter of 840 mm.
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CONCLUSIONS
During the 20th century there has been considerable research into column separation during water hammer or transient events. This report attempts to span all of the significant research that has been carried out during this period. The occurrence of low pressures and associated column separation during water hammer events has been a concern for much of the twentieth century in the design of pipe and water distribution systems. The closure of a valve or shutdown of a pump may cause low pressures during transient events. The collapse of vapor cavities and rejoinder of water columns can generate extremely large pressure that may cause significant damage or ultimately failure of the pipe system. As early as 1900, Joukowsky had identified the physical occurrence of column separation. The 1930s produced the first mathematical models of vapor cavity formation and collapse based on the graphical method. The identification of the various physical attributes of column separation occurred in the mid-20th century (distributed or vaporous cavitation in the 1930s; intermediate vapor cavities in the 1950s). These both led to a better physical understanding of the process of column separation and ultimately laid out the groundwork for the development of computer based numerical models. The late 1960s saw the development of the first computer models of column separation within the framework of the method of
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characteristics solution of the water hammer equations. A variety of alternative numerical models were developed from the late 1960s to the early 1990s. The most significant models that have been developed include: the discrete vapor cavity model (DVCM), the discrete gas cavity model (DGCM) and the generalized interface vapor cavity model (GIVCM). The first two are the easiest to implement. The DVCM is the most popular model used in currently available commercial computer codes for water hammer analysis. The GIVCM handles a number of pipeline configurations (sloping and horizontal pipe) and various interactions between water hammer regions, distributed vaporous cavitation zones, intermediate cavities (along the pipeline) and cavities at boundaries (valve, high point). More accurate treatment of distributed vaporous cavitation zones, shock waves and various types of discrete cavities, contribute to improved performance of the pipe column separation model. Although the interface model gives reliable results, it is quite complicated for general use. The drawback of this model in comparison to the discrete cavity models is the complex structure of the algorithm and longer computational time. At the present stage, the GIVCM is useful as a research tool but not in commercial codes. Numerous experimental studies have been carried out over the last 40 years. From all the validation tests presented in the research literature it may be concluded that, despite its simplicity, the discrete vapor cavity model (DVCM) reproduces the essential features of transient cavitation. The versatility of the model has been demonstrated by the variety of pipe systems used in the tests. The major deficiency of the model is the appearance of non-physical oscillations in the results. The DGCM is recommended in developing and revising industrial engineering water hammer computer codes. Bergant and Simpson (1999a) compared numerical results from discrete vapor (DVCM), discrete gas (DGCM), and generalized interface vaporous cavitation models (GIVCM) with results from measurements. The discrepancies between the measured, DVCM, DGCM and GIVCM results found by temporal and global comparisons may be attributed to approximate modeling of column separation along the pipeline (distributed vaporous cavitation regions, actual number and position of intermediate cavities) resulting in slightly different timing of cavity collapse and superposition of pressure waves. In addition, discrepancies may also originate from discretization in the numerical models, the unsteady friction term being approximated as a quasi-steady friction term and uncertainties in the measurements. In the last 10 years, the amount of research effort into column separation has slowed. There is still room for further research in a number of areas. Laboratory testing of the impact of various cases of distributed cavitation (or vaporous cavitation) in conjunction with the testing of the performance of the three most commonly used models (DVCM, DGCM and GIVCM) needs to be undertaken. Over the last 10 years a number of advances in unsteady friction have occurred. The impact of these new models on the behavior of
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column separation modeling also needs to be investigated. Finally, further work could also be carried out on the GIVCM to explore whether simplifications could be made to the complexity of the approach so that it becomes viable to introduce this approach into commercial water hammer codes.
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ACKNOWLEDGEMENT
The Surge-Net project (www.surge-net.info) is supported by funding under the European Commission’s Fifth Framework ‘Growth’ Programme via Thematic Network “Surge-Net” contract reference: G1RT-CT2002-05069. The authors of this paper are solely responsible for the content and it does not represent the opinion of the Commission. The Commission is not responsible for any use that might be made of data herein.
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Abbreviations in References
ASCE
American Society of Civil Engineers
ASME
American Society of Mechanical Engineers
AWWA
American Water Works Association
BHRA
British Hydromechanics Research Association
BHR Group
British Hydromechanics Research Group
BNES
British Nuclear Engineering Society
FED
Fluids Engineering Division
IAHR
International Association of Hydraulic Research
JSME
Japan Society of Mechanical Engineers
MISI
Moscow Institute of Civil Engineering
PVP
Pressure Vessels and Piping
3R
Rohre Rohrleitungsbau Rohrleitungstransport
SMiRT
Structural Mechanics in Reactor Technology
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8
NOMENCLATURE
Abbreviations DGCM
discrete gas cavity model
DVCM
discrete vapor cavity model
FEM
finite element method
FSI
fluid-structure interaction
GIVCM
generalized interface vaporous cavitation model
MOC
method of characteristics
ODE
ordinary differential equation(s)
Scalars A
cross-sectional pipe area, m2
a
pressure wave speed, m/s
B
known constant in water hammer compatibility equations
C
known constant in water hammer compatibility equations
D
inner diameter of pipe, m
f
Darcy-Weisbach friction factor
g
gravitational acceleration, m/s2
H
(piezometric) head, m
HGL
hydraulic grade line, m
L
pipe length, m
M
mass, kg
N
number of computational reaches
p
fluid pressure, Pa
Q
discharge, m3/s
R
inner radius of pipe, m
Rg
specific gas constant, m2/s2/K
S
cavitation severity index
T
absolute temperature, K
Tc
valve closure time, s
Tcs
duration of the first vapor cavity, s
t
time, s
V
flow velocity, m/s
∀
volume, m3
x
distance along the pipeline, m
z
elevation of centerline of pipe, m
87
α
void fraction
γ
specific weight, kg/m2/s2
∆
numerical step size, change in magnitude
θ
pipe slope, rad
ν
Poisson ratio
ρ
mass density, kg/m3
ψ
weighting factor
Subscripts b
barometric
cs
column separation
f
final
g
gas
in
inception, initial
j
computational section index
k
reach index
m
mixture
max
maximum
mc
negative (minus) characteristic
mi
mixture-inception
mt
mixture-terminal
P
point P in x-t plane
pc
positive (plus) characteristic
q
quasi-steady
r
reversal
RV
reservoir-valve (see Eq. 5)
s
shock
sv
shock-vapor
T
tank
u
upstream, unsteady
v
vapor
vc
vapor cavity
0
initial (steady) state at t = 0
Superscripts *
absolute pressure
88
