154
4 Mine Ventilation Networks
Variable
Q 14
Q 12
Q 24
Q 34
Q 23
F 1
1
−1
0
0
0
F 2
0
1
−1
0
−1
F 3
0
0
0
−1
1
F 4
48
54
48
0
0
F 5
0
0
−48
36
12
The inverse Jacobian matrix is obtained, and the matrix equation of the system
can then be solved:
Q i,0
= −
J i j
−1 [F i ] .
The results of these calculations are shown below:
Solution
0.6190476
0.1904762
0.1428571 0.0079365
0.0039683 Q 14 =
−5.71
−0.380952
0.1904762
0.1428571 0.0079365
0.0039683 Q 12 =
−5.71
−0.190476
−0.404762
−0.303571
0.0039683 −0.008433
Q 24 = −23.48
−0.190476
−0.404762
−0.553571
0.0039683
0.0124008 Q 34 =
17.77
−0.190476
−0.404762
0.4464286 0.0039683
0.0124008 Q 23 =
17.77
The initial, arbitrarily selected, airflow rates are then corrected using the values
of Q ij found in this first iteration. The new, modified airflow rates are then:
Branch
14
12
24
34
23
R ij
0.4
0.3
0.8
0.2
0.1
Q i + Q i
−65.71
84.29
6.52
−72.23
77.77
Using these values, the entire calculation process is repeated and further new
values are obtained for Q ij (1, 2, … n), allowing the system solution to be
approximated.
The numerical results for the second, fourth and fifth iteration are given below.
After the fourth iteration results are the same as two decimal places. Thus, we can
say that convergence is obtained at the fourth iteration.
Iteration 2:
Matrix of the system function (F i )
Function
Q 14
Q 12
Q 24
Q 34
Q 23
Ind. term = Q a
F i
F 1
−65.7
−84.3
0
0
0
150.0 =
0.0
F 2
0
84.3
−6.5
0
−77.8
0 =
0.0
F 3
0
0
0
72.2
77.8
−150.0 =
0.0
F 4
−1727.347 2131.2245 33.985969 0
0
0 =
437.9
F 5
0
0
−33.98597 −1043.496 604.78396 0 =
−472.7
(continued)
4 Mine Ventilation Networks
Variable
Q 14
Q 12
Q 24
Q 34
Q 23
F 1
1
−1
0
0
0
F 2
0
1
−1
0
−1
F 3
0
0
0
−1
1
F 4
48
54
48
0
0
F 5
0
0
−48
36
12
The inverse Jacobian matrix is obtained, and the matrix equation of the system
can then be solved:
Q i,0
= −
J i j
−1 [F i ] .
The results of these calculations are shown below:
Solution
0.6190476
0.1904762
0.1428571 0.0079365
0.0039683 Q 14 =
−5.71
−0.380952
0.1904762
0.1428571 0.0079365
0.0039683 Q 12 =
−5.71
−0.190476
−0.404762
−0.303571
0.0039683 −0.008433
Q 24 = −23.48
−0.190476
−0.404762
−0.553571
0.0039683
0.0124008 Q 34 =
17.77
−0.190476
−0.404762
0.4464286 0.0039683
0.0124008 Q 23 =
17.77
The initial, arbitrarily selected, airflow rates are then corrected using the values
of Q ij found in this first iteration. The new, modified airflow rates are then:
Branch
14
12
24
34
23
R ij
0.4
0.3
0.8
0.2
0.1
Q i + Q i
−65.71
84.29
6.52
−72.23
77.77
Using these values, the entire calculation process is repeated and further new
values are obtained for Q ij (1, 2, … n), allowing the system solution to be
approximated.
The numerical results for the second, fourth and fifth iteration are given below.
After the fourth iteration results are the same as two decimal places. Thus, we can
say that convergence is obtained at the fourth iteration.
Iteration 2:
Matrix of the system function (F i )
Function
Q 14
Q 12
Q 24
Q 34
Q 23
Ind. term = Q a
F i
F 1
−65.7
−84.3
0
0
0
150.0 =
0.0
F 2
0
84.3
−6.5
0
−77.8
0 =
0.0
F 3
0
0
0
72.2
77.8
−150.0 =
0.0
F 4
−1727.347 2131.2245 33.985969 0
0
0 =
437.9
F 5
0
0
−33.98597 −1043.496 604.78396 0 =
−472.7
(continued)