user's manual for corhyd: an internal diffuser hydraulics model - IfH

Fig. 12: Pipe flow after the acceleration of the whole fluid in the outfall took place.
To calculate the time t during accelerations take place estimates using momentum and mass
conservation equations are analyzed for an unsteady, incompressible pipe flow along the
coordinate s following a streamline:
The momentum equation is
1 ∂v
g ∂t + ∂E
∂s = 0 (10)
where E denotes the energy head.
The mass conservation equation for an incompressible fluid (∂ρ/∂t = 0) in an non-deformable
pipe (∂A/∂t = 0) is
∂( ρvA)
∂( + ρA)
= 0 ∂Q
∂s ∂t ∂s = 0 v 1(t)A = v 2 (t)A = Q(t) (11)
Further assuming a pipeline with constant cross section and length L the first term of (10) is
1 ∂v
g ∂t
= 1 dQ O1
g dt ⌡ ⌠ A ds = 1 g
H
dQ
dt
L
A = L g
dv
∂E
dt
and the second term is
∂s = E O - E H + ∆E , where E H (12)
and E O (13) denote the energy heads at the water surfaces at the headworks (E H ) and at the
outlet (E O ) right after the water level rise in the headworks and before acceleration took place
and ∆E the headloss due to friction:
E H = ⎜ ⎛ v H ²
⎝ 2g + p H
γ +z H
⎠ ⎟⎞ = z a + ∆z = z b (12)
E O = ⎜ ⎛ v²
⎝ 2g + p O
γ +z O
⎠ ⎟⎞ = v²
2g + z O,a (13)
∆E = r⎜ ⎛ v²
⎝ 2g⎠ ⎟⎞ where r = λL/D and λ the friction coefficient (14)
(12), (13) and (14) in (10) gives
L dv
g dt + v²
2g + z O,a - z b + r ⎜ ⎛ v²
⎝ 2g ⎠ ⎟⎞ = 0 (15)
Additionally, for the terminal velocity v b it is
z O,a - z b = -(1+r) ⎜ ⎛ v b ²
⎝ 2g⎠ ⎟⎞
(16)
Institut für Hydromechanik, Universität Karlsruhe 25