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

3.3 Solving scheme
The governing equation can be solved either for a given head or a given total discharge. For
both a first estimate is used as a starting value and further iterations lead to the final value.
3.3.1 Solving for total head
At the first port/riser on the seaward side (i = 1) an initial discharge q 1 is estimated, for
example q 1 = Q/N with Q = total discharge and N = total number of risers. Equation (19) then
allows to calculate the first internal pressure of the diffuser p d,1 . The further discharges q 2
until q N are calculated using equation (20). A final application of equation (18) allows to
calculate p d,N+1 , the necessary pressure at the headworks to drive the system. The total head H t
can be calculated by H t = p d,N+1 /γ effluent if the water level elevation of a gravity driven system
has to be defined. The calculated total discharge is Q c =∑
N
q k
k = 1
. The difference to the planned
total discharge is diffc = Q - Q c . If necessary (i.e. for diff c > Q/10000) CorHyd performes
further iterations with modified estimates q 1,c .
To achieve faster convergence the following algorithm (eq. (22)) has been implemented to
calculate q 1,c :
q 1,1 = Q/N; q 1,2 = q Q 1,1 ; Q
q1,c = q 1,c-2 diff c-1 -q diff
1,c-1 c−2
for (c>2) (22)
1
diff
1
− diff
2
The iteration stops if the difference between the given total discharge and the calculated total
discharge is less than 10 -5 Q. The results are individual port/riser discharges and velocities in
all pipe sections along the diffuser and a total head. These can be displayed or printed with
further output options.
c−
c−
3.3.2 Solving for total flow
At the first port/riser on the seaward side (i = 1) an initial internal pressure p d,1 is estimated,
for example p d,1 = H t γ e /N + p a,1 + γ e (z jet,i - z d,i ) with H t = total head at headworks. Equation
(19) then allows to calculate the first discharge q 1 . The further discharges q 2 until q N are
calculated using equation (20). A final application of equation (18) allows to calculate p d,N+1 ,
the necessary pressure at the headworks to drive the system. The total head H t can be
calculated by H t = p d,N+1 /γ e if the water level of a gravity driven system has to be defined. The
N
q k
k = 1
calculated total discharge is Q c =∑
. The difference to the planned total head is diffc = H t -
H tc . If necessary (i.e. for diff c > H t /10000) CorHyd performes further iterations with modified
estimates p d,1,c .
To achieve faster convergence the following algorithm has been implemented to calculate
p d,1,c :
p d,1,1 = H t γ e /N+p a,1 +γ e (z jet,i -z d,i ); p d,1,2 = p H t
d,1,1 ; H
pd,1,c = p d,1,c-2 diff c-1 -p diff
d,1,c-1 c−2
t1
diffc−
1
− diffc−
2
for (c>2) (23)
Institut für Hydromechanik, Universität Karlsruhe 31