Rigid Column ModelThe concept of the rigid column model is based on neglecting the water compressibility so that the velocity and pressure are space-invariant variables throughout the flow, which is assumed to be a rigid water column. Following other references, e.g., Hatcher et al. (2015), Vasconcelos and Leite (2012), Vasconcelos et al. (2011), and Zhou et al. (2002), using the rigid column model, the air–water interface was assumed to be vertical so that the free-surface flow zone was neglected. In addition, the pressure head throughout the pressurized flow was assumed to be constant and equal to the reservoir pressure head. Thus, the governing equations include the momentum equation of the water column [Eq. (2)], the continuity [Eq. (3)], and the time derivative of the polytropic process relationship of the ideal gas law [Eq. (4)], applied to the air-pocket. In most of the relevant studies, the air-pocket was simulated using the thermodynamic law, so that the air velocity was neglected (2) dQdt=gAL[Hres−(Hair−Hatm)−((fLD+Kloss)Q|Q|2gA2)](3) (4) where t = time variable; Q = water discharge; Va = air-pocket volume; Hair = air absolute pressure head; Hatm = atmospheric absolute pressure head, which was set to 10.33 m; L = equivalent water column length, calculation of which is explained subsequently; A = cross-sectional area of pipe; Kloss = summation of local losses; and k = polytropic coefficient. Following Lee (2005), the polytropic coefficient k was set to k=γ=1.4, where γ represents the adiabatic constant of the air. Thus, the air-pocket was assumed to undergo an adiabatic process, in which the air pressure, volume, and temperature changed by air compression and expansion. The air volume and temperature variations can be realized from the results of the air pressure variations that are presented subsequently.The equivalent water column length is calculated as (5) to compensate for the water volume of the free-surface flow, which is neglected in applying the rigid column model.The set of Eqs. (2)–(4) of the rigid column model was solved using the classical fourth-order Runge–Kutta method, as described in the literature (e.g., Rokhzadi and Fuamba 2020; Press et al. 2007).Method of Characteristics ModelEq. (6) is the governing equations set for one-dimensional flow in closed conduits. These equations constitute a pair of partial differential equations (PDEs), which are the momentum and continuity equations, respectively (Wylie and Streeter 1993; Chaudhry 2014) (6) ∂V∂t+g∂H∂x+f|V|V2D=0∂H∂t+a2g∂V∂x=0where V = water velocity; H = piezometric head of pressurized flow; a = acoustic wave speed; and x = spatial variable along pipe axis, with the positive direction from the upstream to the downstream.The method of characteristics is a numerical method with first-order accuracy, which commonly has been used to solve the water hammer equations. The method of characteristics allows transforming the pair of PDEs [Eq. (6)] into two ordinary differential equations (ODEs) along two positive and negative characteristics. Further details of this method are available in the literature (e.g., Wylie and Streeter 1993).For the first node of the pressurized flow zone at the upstream (the one near the reservoir), only the negative characteristic C−, which originates somewhere between the first and second nodes, can be used for the calculation. Thus, for the first node, the energy equation between the reservoir and the first node of the pressurized zone was used instead of the equation along the positive characteristic (7) H1n+1=(Hres+Hatm)−(1+Kloss)×Q1n|Q1n|2gA2where subscript 1 denotes first node of pressurized flow zone.For the last node of the pressurized flow zone (near the air-pocket), only the equation along the positive characteristic C+ can be used. Therefore, instead of the negative characteristic line, the energy equation between the last node of the pressurized zone and the air-pocket was used (8) HNn+1=Hairn+1−QNn+1|QNn+1|2gA2where N denotes last node of pressurized flow zone.The air-pocket pressure also was calculated using the polytropic process relationship of the ideal gas law.Friction Factor AnalysisIn this paper, an extra friction factor is proposed to calculate properly the energy dissipation, which cannot be predicted using only the steady friction factor. Thus, the friction term in the governing equations is (9) Following Lee (2005), the effective parameters of a transient flow in a reservoir–pipe system with one end blocked and followed by air entrapment could be (10) F(V,Va,Pres,P0,La,L,aa,a,ρa,ρ,ϑa,ϑ,τw,D,ε)=0where Va = air velocity; Pres and P0 = absolute pressures of reservoir and atmosphere, respectively; La and L = air and water lengths, respectively; aa and a = acoustic wave speeds of air and water phases, respectively; ρa and ϑa = air density and kinematic viscosity, respectively; ρ and θ = water density and kinematic viscosity, respectively; τw = wall shear stress; and ε = nondimensional roughness coefficient, which depends on the pipe’s material property. The physical properties of the air phase, including ϑa and aa, are significantly less than those of water. Furthermore, because the air phase is modeled as a lumped gas, in which the air dynamic including Va is neglected, Eq. (10) can be simplified as (11) F(V,Pres,P0,La,L,a,ρa,ρ,ϑ,τw,D,ε)=0Eq. (11) considers the effect of air density because, based on the literature, the air mass is an influential parameter.In the Buckingham π theorem of dimensional analysis, the recurring set chosen in this study was L, ρ, and V. Thus, because there are 12 variables in Eq. (11) and there are three fundamental dimensions—mass, time, and length—9 nondimensional variables can be obtained. Therefore, Eq. (11) can be rearranged in a nondimensional form as (12) τwρV2=F(LaL,ρaρ,PresρV2,P0ρV2,DL,aV,ε,ϑLV)The term on the left-hand side of Eq. (12) implies the friction factor, and the last term on the right-hand side implies the inverse of the Reynolds number. In addition, the last two terms on the right-hand side can be represented by the steady-state friction factor. Following Eq. (12), a formula for the additional friction factor can be proposed. In this study, to include all parameters influencing the friction factor and to involve the important physical parameters, the additional friction factor is introduced as (13) f′=f′(PresP0,ρaρ,LaL,agD,f)Zhang et al. (2018) carried out the dimensional analysis of the additional friction factor for the full pipe flow and found a linear relationship with respect to the ratio of wave speeds and the ratio of air mass to water mass. Thus, following Zhang et al. (2018), a similar relationship was considered in this paper as follows (14) f′=C′×PresP0×ρaρ×LaL×agD×fA more precise formula can be found by running experiments and using the curve-fitting technique. Therefore, the following formula is proposed: (15) f′=C′×Hres+HatmHatm×ρaρ×LaL×agD×fwhere C′ = calibration factor. Following Wan et al. (2010) and Zhang et al. (2018), f can be calculated as f=(64ν)/(VD) for laminar flow regimes. In addition, for turbulent flow regimes, f=(8gnM2)/(Rp3), where nM is the roughness coefficient, and Rp is the hydraulic radius. Therefore, a compound model can be used depending on the critical Reynolds number (Rc) (16) f={64νVDR≤Rc (laminar)8g(nM)2Rp3R>Rc (turbulent)where Rc=2,320 (Eckhardt 2009); and Rp for circular pipe can be calculated as D/4. The steady-state friction factor for turbulent flow is a function of the roughness parameter only [Eq. (16)]. This implies that when a turbulent flow occurs, it is assumed to be a fully turbulent flow, which is more relevant for transient flows in SWS. However, any other formula in which R is involved also can be used.During the simulation, when V=0, because the coefficient f is not defined, the additional friction factor f′ is set to zero.To facilitate the optimization procedure, C′ is replaced with C×103, and the coefficient of C is moved to the denominator of the ratio ρa/ρ. Thus (17) f′=C×Hres+HatmHatm×ρa10−3ρ×LaL×agD×λFollowing Hatcher and Vasconcelos (2017) and Wylie and Streeter (1993), the acoustic wave speed in a conduit containing water and air is calculated as (18) a=K/ρ1+KD/Ee+mRT(K/P0−1)/P0where R = gas constant (8.314  J·K−1·mol−1); T = absolute temperature (293 K); m = air content; K = volume modulus (2.15 GPa); P0=ρgHatm = atmospheric absolute pressure; e = thickness of PVC pipe (7 mm); E = elasticity modulus of PVC pipe (2.5 GPa); and ρ = liquid density (998  kg/m3).To take into account the effect of the additional friction factor, in the governing equations in the previous subsections, f is replaced with f+f′. The optimum C value, proposed for different ranges of air-pocket size (small and large), is calculated as explained in the following paragraph.The governing equations were solved for different C values, and the pressure distributions calculated by both the rigid column and the method of characteristics models were compared with the relevant experimental data, and the sum of squared residuals (S) was calculated as (19) S=∑i=1np([hpe(i)−hp(i)]2+[hve(i)−hv(i)]2)where np = number of peak values; hpe, and hve = experimental positive and negative peaks, respectively; and hp, and hv = numerical positive and negative peaks, respectively. Finally, the value of the calibration factor (C) that corresponds to the minimum S value is considered to yield the optimized additional friction factor (f′). The results of this optimization are presented in the next section. The additional friction factor (f′) is calculated in a similar way for both the rigid column and the method of characteristics models.

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