Celerity and Pipe Elasticity
The elasticity of any medium is characterized by the deformation of the medium due to the application of a force. If the medium is a liquid, this force is a pressure force. The elasticity coefficient (also called the elasticity index, constant, or modulus) is a physical property of the medium that describes the relationship between force and deformation.
Thus, if a given liquid mass in a given volume (V) is subjected to a static pressure rise (dp), a corresponding reduction (dV < 0) in the fluid volume occurs. The relationship between cause (pressure increase) and effect (volume reduction) is expressed as the bulk modulus of elasticity (En) of the fluid, as given by:
bulk modulus of elasticity
static pressure rise
incremental change in liquid volume with respect to initial volume
incremental change in liquid density with respect to initial density
A relationship between a liquid's modulus of elasticity and density yields its characteristic wave celerity:
The characteristic wave celerity (a) is the speed with which a disturbance moves through a fluid. Its value is approximately 4,716 ft./sec. (1,438 m/s) for water and approximately 1,115 ft./sec. (340 m/s) for air.
Injecting a small percentage of small air bubbles can lower the effective wave speed of the fluid/air mixture, provided it remains well mixed. This is difficult to achieve in practice, because diffusers may malfunction and air bubbles may come out of suspension and coalesce or even buoy to the top of pipes and accumulate at elbows, for example.
In 1848, Helmholtz demonstrated that wave celerity in a pipeline varies with the elasticity of the pipeline walls. Thirty years later, Korteweg developed an equation to determine wave celerity as a function of pipeline elasticity and liquid compressibility. Bentley HAMMER V8i Edition uses an elastic model formulation that requires the wave celerity to be corrected to account for pipeline elasticity.
Equation is valid for thin walled pipelines (D/e > 40). The factor y depends on pipeline support characteristics and Poisson's ratio. y depends on the following:
- Pipe is anchored throughout against axial movement: y = 1 µ2, where µ is Poisson's ratio
- Pipe is equipped with functioning expansion joints throughout: y = 1 µ/2
- Pipe is supported only at one end and allowed to undergo stress and strain both laterally and longitudinally: y = 5/4 m (ASCE, 1975)
For thick-walled pipelines, various theoretical equations have been proposed to compute celerity; however, field investigations are needed to verify these equations. Tables Table 14-2: Physical Properties of Some Common Pipe Materials and Table 14-3: Physical Properties of Some Common Liquids provide values for various pipeline materials and liquids that are useful to calculate celerity during transient analysis. Figure 14-6: Celerity versus Pipe Wall Elasticity for Various D/e Ratios provides a graphical solution for celerity given pipe-wall elasticity and various diameter/thickness ratios.
Table 14-2: Physical Properties of Some Common Pipe Materials Material Young's Modulus Poisson's Ratio, (109 lbf/ft2) (GPa) Steel 4.32 207 0.30 Cast Iron 1.88 90 0.25 Ductile Iron 3.59 172 0.28 Concrete 0.42 to 0.63 20 to 30 0.15 Reinforced Concrete 0.63 to 1.25 30 to 60 0.25 Asbestos Cement 0.50 24 0.30 PVC (20oC) 0.069 3.3 0.45 Polyethylene 0.017 0.8 0.46 Polystyrene 0.10 5.0 0.40 Fiberglass 1.04 50.0 0.35 Granite (rock) 1.0 50 0.28
Table 14-3: Physical Properties of Some Common Liquids Liquid Temperature (oC) Bulk Modulus of Elasticity Density (106 lbf/ft2) (GPa) (slugs/ft3) (kg/m3) Fresh Water 20 45.7 2.19 1.94 998 Salt Water 15 47.4 2.27 1.99 1,025 Mineral Oils 25 31.0 to 40.0 1.5 to 1.9 1.67 to 1.73 860 to 890 Kerosene 20 27.0 1.3 1.55 800 Methanol 20 21.0 1.0 1.53 790
Figure 14-6: Celerity versus Pipe Wall Elasticity for Various D/e Ratios
For pipes that exhibit significant viscoelastic effects (for example, plastics such as PVC and polyethylene), Covas et al. (2002) showed that these effects, including creep, can affect wave speed in pipes and must be accounted for if highly accurate results are desired. They proposed methods that account for such effects in both the continuity and momentum equations.
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