By Gilberto E. Urroz, September 2004 NOTE: The formulations shown in this document are taken from Tullis, J.P., 1989 , Hydraulics of Pipelines – Pumps, Valves, Cavitation, Transients, John Wiley & Sons, New York Hydraulic Transients Hydraulic transients in pipelines occur when the steady-state conditions in a given point in the pipelinestart changing with time, e.g., shutdown of a valve, failure of a pump, etc. In order to account for the disturbance in the steady-state conditions, a pressure wave will travel along the pipeline starting at the point of the disturbance and will be reflected back from the pipe boundaries (e.g., reservoirs) until a new steady-state is reached. The pressure waves in pipelines travel at a constantcelerity as described next. Celerity of pressure waves in pipelines A pressure disturbance in a pipeline propagates with a very large wave celerity a given by K a=
C⋅K ⋅D 1+ E ⋅ ∆D
where K and ρ are the bulk modulus of elasticity and density of the fluid, D and ∆D are the inner diameter and the thickness of the pipe, E is the Young modulus (modulus of elasticity) of the pipe material,and C is a coefficient that accounts for the pipe support conditions: • • • C = 1 – 0.5µ, if pipe is anchored at the upstream end only C = 1-µ2, if pipe is anchored against any axial movement C = 1, if each pipe section is anchored with expansion joints at each section
Here, µ represents the Poisson’s ratio of the material. For a perfectly rigid pipe, E is infinite, and the wave celeritysimplifies to
Equations of hydraulic transients The equations governing hydraulic transients are a pair of coupled, non-linear, first-order partial differential equations, namely,
∂H fV | V | ∂V ∂V + +V + =0 ∂x 2D ∂x ∂t
V ∂H ∂H a 2 ∂V = 0, + + ∂t ∂x g ∂x
where H is the piezometric head (=z+p/g), V is the flow velocity, x is thedistance along the pipe, t is time, g is the acceleration of gravity, f is the pipe friction factor (assumed constant), D is the pipe diameter, and a is the celerity of a pressure wave in the pipeline. These equations can be simplified by recognizing that the advective terms V∂V/∂x and V∂H/∂x, are negligible when compared to other terms in their corresponding equations. The resulting equations are,thus, g ∂H fV | V | ∂V + + =0 ∂x 2D ∂t
∂H a 2 ∂V =0 + ∂t g ∂x
The solution of the two transient equations require us to find the values of H(x,t) and V(x,t) in a domain defined by 0 < x < L, and 0 < t < tmax, where L is the length of the pipe, and tmax is an upper limit for the time domain. While the simultaneous solution of these two equations is possible throughthe use of finite differences, in this document we will approach the solution by determining the characteristic lines of the problem.
Characteristics in pipe transients The characteristic lines for this case will be lines in the x-t plane defined by dx/dt = u(x,t), along which a function of H and V is conserved in time. Since the momentum and continuity equations shown above include partialderivatives of H and V with respect to x and t, we will try to reconstruct out of them expressions for the total derivatives of H and V, namely, dH/dt = (∂H/∂x)(dx/dt)+ ∂H/∂t, and dV/dt = (∂V/∂x)(dx/dt)+ ∂V/∂t.
We start by adding the momentum equation to the continuity equation multiplied by a parameter λ, i.e.,
g ∂H a 2 ∂V ∂H fV | V | ∂V + + + λ ⋅ ∂t + g ∂x = 0 . 2D ∂x ∂t Expanding this equation and collecting terms that resemble the total derivatives dH/dt and dV/dt, produces:
2 ∂H g ∂H ∂V λa ∂V fV | V | + + =0 + + g ∂x 2D ∂t λ ∂x ∂t
The terms between parentheses in this equation correspond to dH/dt and dV/dt, respectively, if we take dx/dt = g/λ for dH/dt and dx/dt = λa2/g for dV/dt. Since the two expressions for...