4.10 Complex Networks
139
(continued)
Using Kirchhoff’s first law:
Q 23 =
87.10
Q 14
−69.30
Q 12
80.70
X 2 =
0.00
Q 24
−6.40
Y 2 =
0.00
Q 34
−62.90
Z = X 2 + Y 2 =
0.00
Q 23
87.10
−Q 24
6.40
The above calculations were made using the Excel Solver to find the value 0. The total power of
the net is 406.9 kW.
4.10.3 Hardy–Cross Method
The Hardy–Cross method (Cross 1936) is a numerical calculation procedure that has
its origin in the field of structural engineering. It is an iterative method developed to
solve calculations for fluid flow networks. The method has the advantage of being
self-correcting and will converge even where initial conditions are poorly specified.
Demonstration
Taking a mesh for which the flows of each of the branches have been estimated, let
Q i be the initial airflow rate assigned to the ith branch of the network and where Q
is the correction made to the flow rate after initial calculations Then, the corrected
flow rate will be:
Q = Q i + Q
This new value of flow rate, Q, becomes the value Q i for the following iteration
and Q is the positive or negative correction that must be made in all branches of
the mesh. Thus, for the next iteration, the pressure drop for one branch (P i ) will
be:
P i = R i (Q i + Q)
2
That can be written as:
P i = R i Q
2
i
1 +
Q
Q i
2
By expanding the binomial, we get:
139
(continued)
Using Kirchhoff’s first law:
Q 23 =
87.10
Q 14
−69.30
Q 12
80.70
X 2 =
0.00
Q 24
−6.40
Y 2 =
0.00
Q 34
−62.90
Z = X 2 + Y 2 =
0.00
Q 23
87.10
−Q 24
6.40
The above calculations were made using the Excel Solver to find the value 0. The total power of
the net is 406.9 kW.
4.10.3 Hardy–Cross Method
The Hardy–Cross method (Cross 1936) is a numerical calculation procedure that has
its origin in the field of structural engineering. It is an iterative method developed to
solve calculations for fluid flow networks. The method has the advantage of being
self-correcting and will converge even where initial conditions are poorly specified.
Demonstration
Taking a mesh for which the flows of each of the branches have been estimated, let
Q i be the initial airflow rate assigned to the ith branch of the network and where Q
is the correction made to the flow rate after initial calculations Then, the corrected
flow rate will be:
Q = Q i + Q
This new value of flow rate, Q, becomes the value Q i for the following iteration
and Q is the positive or negative correction that must be made in all branches of
the mesh. Thus, for the next iteration, the pressure drop for one branch (P i ) will
be:
P i = R i (Q i + Q)
2
That can be written as:
P i = R i Q
2
i
1 +
Q
Q i
2
By expanding the binomial, we get: