VIRUS QUE AFECTAN AL CULTIVO DE LA PATATA

Various mechanism can be responsible for the observed N1 values. Clayton and van Zyl (2007) investigated four factors that may be responsible for the range of leakage exponents that have been observed in the field and in experimental studies, namely (a) leak hydraulics, (b) pipe material behaviour, (c) soil hydraulics, and (d) water demand. These factors will be discussed here in some detail.

(a) Leak hydraulics: The hydraulic behaviour of orifices has been studied extensively and can

be predicted with a great degree of certainty. It has been accepted that the leakage exponent of a fixed leak can be assumed to be 0.5 and that the discharge coefficient is often not constant but expressed in terms of the Reynolds number. It is therefore feasible to assume that a certain type of leak can be modelled using a fixed discharge coefficient, but with varying leakage exponents (i.e. higher or lower than the theoretical 0.5).

Another aspect of leak hydraulics that can affect the leakage exponent N1 is the flow regime, whether it is turbulent or laminar. Experiments have shown the typical values of N1 for laminar, turbulent and transitional flow conditions:

Table 2-6: The values of N1 for different flow regimes (Clayton & van Zyl, 2007)

Flow regime Re N1

Laminar flow < 10 1

Transitional flow 0.5 - 1

Fully turbulent flow >4000-5000 0.5

If the Reynolds number is written as a function of discharge through the leak, the formula is as follows:

𝑅_𝑒 =4𝑣𝐴 𝑘𝑃

Equation 2-8

Where v is the velocity (m/s), P is the wetted perimeter (m) , Re is the Reynolds number, and

k = Kinematic viscosity (m2/s). Two expressions can be developed to estimate the maximum laminar and transitional flow rates that are possible in a typical water distribution system:

2-42 𝑞 = 𝜋𝑣 2_𝑅 𝑒 2 4𝐶𝑑√2𝑔ℎ Equation 2-9 𝑞 =(𝑛 + 1)𝑘 2_𝑅 𝑒2 4𝐶_𝑑𝑛√2𝑔ℎ Equation 2-10

Where h is the pressure head (m), g is the acceleration due to gravity (m/s2), n is the aspect ratio of a rectangle, and Cd is the discharge coefficient. Using equations 2-6 and 2-7, Clayton & van

Zyl (2007) plotted leak flow rate (l/h) against typical pressure range (m) for different types of leak openings. Figure 2-14 shows the results of this plot. The maximum laminar and transitional flow rates for the different leak types are illustrated.

Figure 2-14: The maximum laminar and transitional flow rates for different leak openings (Clayton & van Zyl, 2007)

Figure 2-14 shows that certain types of leak openings (namely rectangular) have higher laminar and transitional flow rates than round holes and square holes. This is due to their much larger wetted perimeters.

(b) Pipe material behaviour: The material behaviour of pipes is thought to play the most

significant role in affecting the pressure-leakage relationship and thus the N1. Understanding the failure behaviour and the associated leakage exponents will assist in leakage modelling and

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in understanding the complexity. It has become more apparent that leak areas are not fixed but expand, to varying degrees, with increasing pressure. Some leak areas also remain closed at low pressures but open when the pressure is increased high enough.

Buckley, (2005) carried out theoretical work at the University of Johannesburg’s Water Research Group of which he developed the following basic model for the flow rate through a round hole in an elastic pipe taking into account the effect of pipe material expansion on the leakage rate. The developed model was expressed as follows:

𝑄 = 𝐶𝑑 𝜋𝑑₀2 4 √2𝑔 (𝐻 0.5₊2𝑐𝜌𝑔𝐷 3𝑡𝐸 𝐻 1.5₊𝑐 2_𝜌2_𝑔2_𝐷2 9𝑡2_𝐸2 𝐻2.5) Equation 2-11

Where d0 is the Original Leak Hole Diameter, D is the Pipe Diameter, E is the Elastic Modulus,

t is the Pipe Thickness and c is a Constant. From Equation 2-8, it is interesting to note that the

leakage exponents vary from 0.5 to 2.5; this corresponds to the N1 values observed in the field and in experimental tests (Clayton & van Zyl, 2007). This relationship also shows that the leak area expansion is much more complex than the power equation indicates.

When leakage was calculated using equation 2-8, it was found that the terms with the exponents of 1.5 and 2.5 contribute very little to the leakage rate under normal pressure conditions for round holes. Hikki (1981) supports this phenomenon that the leakage through round holes can be characterised by a leakage exponent of 0.5.

Another aspect of pipe material that can affect the N1 is the material property, because pipes fail in different characteristic ways depending on the type of pipe, as shown in .

Table 2-7. This table indicates that the leakage exponents (N1) varies for different materials and different failure types.

Table 2-7: The leakage exponents for various pipe materials (Greyvenstein & van Zyl, 2005)

Failure type Leakage exponent

uPVC Asbestos cement Mild steel

Round hole 0.52 - 0.52 Longitudinal crack 1.38 – 1.85 0.79 – 1.04 - Circumferential crack 0.41 – 0.53 - - Corrosion cluster - - 0.67-2.30

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There are some key insights that can be drawn from Table 2-7. Firstly, the highest leakage exponents were found to occur in corroded steel pipes. Secondly, it was found that round holes generally have leakage exponents close to the theoretical 0.5, and that this is true regardless of the pipe material. Thirdly, besides the high exponents resulting from corrosion failure, longitudinal cracks were found to also have high leakage exponents. Finally, the leakage exponent of the circumferential crack was found to be less than 0.5, implying that the crack could be closing up as the internal pressure increases.

Clayton and van Zyl's (2007) study also confirmed these findings and discussed the fact the different materials fail in different characteristic ways. They also mention that longitudinal cracks are common in asbestos cement, while metallic pipes such as steel and cast-iron pipes often leak through holes formed by corrosion. Small cast iron pipes were found to fail in bending, resulting in circumferential cracks. Due to the high coefficient of thermal expansion these cracks open and close depending on the temperature in small cast-iron pipes.

Ssozi, Reddy and van Zyl (2015) carried out a finite element study in which they investigated the viscoelastic behaviour of pipe materials. The study found that materials that undergo viscoelastic behaviour generally have higher N1 exponent values as shown in Figure 2-15 below.

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(c) Soil hydraulics: Another factor that plays a part in the pressure-leakage relationships,

observed in the field and in laboratory experiments, is the effect of soil around a leaking pipe. Water that leaks from the pipe travels through the surrounding soil. Following a simplistic model of geotechnical seepage theory, the leakage flow rate should be linearly proportional to the pressure head of the water in the pipe. This is also known as Darcy’s Law and is expressed as follows:

𝑄 = 𝐹 𝑘 ℎ

Equation 2-12

Where q is leakage flow rate from the pipe, 𝐹 is the shape factor for the soil, k is the coefficient of permeability of the soil and ℎ is the pressure head. Clayton and van Zyl (2007) mention the fact that the assumptions that underpin equation 2-9 do not necessarily apply to leaking pipes. One of the reasons for this is that the interaction between the surrounding soil and leaking pipe is a complex phenomenon as shown in Figure 2-16. It can be seen from the figure that a fluidised zone develops when the water jet interacts with the soil particles.

Figure 2-16: Fluidisation zone from vertical water leak jet (Ma, 2011)

This phenomenon was investigated further by Pike (2015) in an experimental study

. In his study it was demonstrated that due to the scouring process that occurs at the pipe wall, small leaks have the potential to develop into large leaks over time, as shown in Figure 2-17. This process was found to be highly influenced, mainly by the orientation of the leak but also by the size of the soil grains and high flow rates.

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Figure 2-17: The initial orifice condition (left) and visual inspection after 100 hours of exposure to scouring (right) (Pike, 2015)

As a result of the interaction between the soil mass and water jet from the leaking pipe, the relationship between head loss and flow is unlikely to be linear. In addition, the turbulent flow regime, the changing geometry of the unconfined flow regime, hydraulic fracturing and piping all contribute towards a complex interaction between the soil particles and the flow rate from the leak (Clayton & van Zyl, 2007).

(d) Water demand: The aspect of water demand is also thought to play an influential role in

the pressure-leakage relationship observed in the field and in experimental tests. During field tests it is often impossible to distinguish between legitimate water consumption and system leakage. For this reason it is pivotal to understand the relationship between pressure and legitimate water consumption. Clayton and van Zyl (2007) express the effect of pressure on water demand as follows:

𝑄𝑑𝑒𝑚𝑎𝑛𝑑 = 𝐶ℎ𝛽

Equation 2-13

Where Qdemand is the legitimate water demand, C is a constant coefficient and 𝛽 is the demand

elasticity to pressure. It is quite clear that equation 2-10 resembles the N1 leakage equation. The demand elasticity takes into account human behaviour change, e.g. with increased water pressure through taps as is illustrated below.

Bartlett, (2004) conducted a study on the water consumption at a student village at the University of Johannesburg. In his study, Bartlett (2004) varied the system pressure of the

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village and monitored the associated water demand. He found that the demand elasticity for indoor usage was approximately 0.2. The outdoor water demand elasticity was found to be 0.5. Outdoor consumption was found to be time based rather than volume based, indicating that higher exponents are expected for outdoor water demand.

In large systems it is inevitable to include legitimate water consumption in the minimum measured night flows. Since the combined leakage exponent can be found to be less than 0.5, Clayton and van Zyl (2007) claim that there is a possibility that system leakage exponents are underestimated if they are measured in systems that contain demand.