Hydraulic Reliability Diagram and Network Buffer Index as Indicators of Water Distribution Network Resilience - Pattern Recognition for Reliability Assessment of Water Distribution Networks - page 108

In case of failure of any of the two pipes, the additional capacity will be taken over by the
remaining pipe, increasing the head loss i.e. reducing the head and pressure in the discharge
node ( H n and p n /ρg , respectively). Thus, in case of failure of pipe 1:

p − p

n n , f 1 2 2

2

H − H = = R Q − R Q

5.3

n n , f 1 2 , f 1 2 , f 1 2

ρ g

Parameter R 2,(f 1 ) stands for the resistance of pipe 2 before and after the failure of pipe 1,
respectively, which by applying the Darcy-Weisbach formula is calculated as:

8 λ L

2 , f 1 2

R =

5.4

2 , f 1

g π

2

D

5
2

taking into consideration the pipe length L 2 , diameter D 2 and the friction factor λ 2,(f 1 ) before
and after the failure of pipe 1. Furthermore, for Q tot = Q 1 + Q 2 Equation 5.3 transforms into:

∆ p R

1

( ) 2

n , f 1 2 , f 1 2

= Q − Q − Q 5.5

2 , f 1 tot 1

R ρ g R

2 2

and finally, for:

∆ p R λ

1 n , f 1 2 , f 1 2 , f 1

= b ; = = a 5.6

R ρ g R λ

2 2 2

Equation 5.5 can be rewritten in the format of Equation 5.2:

 

2

aQ − b

 

2 , f 1

Q = Q 1 − 5.7

 

1 tot

Q



tot



Having the available demand fraction in case of the failure of pipe 1 as ADF 1 = Q 2,f 1 /Q tot ,
Equations 5.2 and 5.7 become identical in case of parameters a = 1 and b = 0.

The condition a = 1 is satisfied in case of flow in the zone of developed (i.e. rough)
turbulence where the condition Q 2,f 1 > Q 2 still yields λ 2,f 1 = λ 2 . In the zone of transitional
turbulence, factor a (actually its squared root) would significantly deviate from 1 only in case
of significant pipe flow increase resulting from the failure. The Moody diagram in Figure 5.7
shows that in the most extreme cases, a doubling of pipe flow (the Reynolds number increase
from, say, 10,000 to 20,000) for extraordinary low values of k/D (say, 0.000001 representing
very big and smooth pipes) would result in the increase of λ up to 20 %; this is for instance a
situation where a 1000 mm pipe would carry flow in order of only 10 l/s, which is not very
common in practice. Much more likely is that significant flow increase will occur in pipes of
smaller diameters (i.e. with higher k/D ) which then affects the friction factor not more than a
few percent. Therefore, a ≈ 1 for flows in the zone of transitional turbulence in most of
situations that normally occur in practice.