132
4 Mine Ventilation Networks
Solving the system, we have:
Q 12 = 65.47 m
3 s
−1 (515.5 m
3 s
−1 is discarded because it has no physical meaning)
Q 14 = 150 − Q 12 = 84.53 m
3 s
−1
Q 23 = Q 12 + 50 = 115.47 m
3 s
−1
Q 34 = Q 12 − 100 = −34.53 m
3 s
−1
Discussion
The airflow rate in branch R 34 (Q 34 = −34.53 m
3 s
−1 ) is in the opposite direction
to that initially assumed. Thus, we can conclude that sign given to the monomial R ij
Q
2
ij was incorrect. Since the initial directions of the airflows within the network are,
in principle, unknown, this is a very important problem to take into account.
Alternative resolution: Methodology employing absolute values
The approach is the same as described above but the monomial R ij Q
2
ij derived using
Kirchhoff ’s second law is written in the form R ij ǀQ ij ǀ Q ij . This formulation makes
the equation robust to any initial choice of circulation direction.
In this way, alternative expressions for the pressure changes in M 1 and M 2 are
obtained which are summarized in the tables below. Once again the equation for M 2
is multiplied by (−1) and summed with that for M 1 to eliminate the variable P v .
Mesh 1 (M 1 )
Corrected approach
+R 12 Q 2
12
+R 12 Q 12 |Q 12 |
−R 24 Q 2
24
−R 24 Q 24 |Q 24 |
−R 14 (Q e − Q 12 ) 2
−R 14 (Q e − Q 12 ) |Q e − Q 12 |
(−1) · Mesh 2 (M 2 )
Corrected approach
+R 23 (Q 12 + Q 24 ) 2
+R 23 (Q 12 + Q 24 ) |Q 12 + Q 24 |
+R 34 ((Q 24 − Q e ) + Q 12 ) 2
+R 34 (Q 24 − Q e + Q 12 ) |Q 24 − Q e + Q 12 |
+R 24 Q 2
24
+R 24 Q 24 |Q 24 |
Unlike the previous method, here, we end up with an equation effectively
containing two unknowns, the variable Q i and the absolute value of Q i . This, of
course, has no direct solution. To get over this difficulty, we use the Solver Solution
Search System from Excel for its simplicity. Entering the values from the problem
into the equation, we have found from the mesh analysis and imposing the condition
that Q 12 ≤ Q e , (that is Q 12 ≤ 150) at all times, that:
Mesh 1
Mesh 2
M 1 + M 2
f (solver):
−3216.83
3216.83
0.00
Q 12
68.80
4 Mine Ventilation Networks
Solving the system, we have:
Q 12 = 65.47 m
3 s
−1 (515.5 m
3 s
−1 is discarded because it has no physical meaning)
Q 14 = 150 − Q 12 = 84.53 m
3 s
−1
Q 23 = Q 12 + 50 = 115.47 m
3 s
−1
Q 34 = Q 12 − 100 = −34.53 m
3 s
−1
Discussion
The airflow rate in branch R 34 (Q 34 = −34.53 m
3 s
−1 ) is in the opposite direction
to that initially assumed. Thus, we can conclude that sign given to the monomial R ij
Q
2
ij was incorrect. Since the initial directions of the airflows within the network are,
in principle, unknown, this is a very important problem to take into account.
Alternative resolution: Methodology employing absolute values
The approach is the same as described above but the monomial R ij Q
2
ij derived using
Kirchhoff ’s second law is written in the form R ij ǀQ ij ǀ Q ij . This formulation makes
the equation robust to any initial choice of circulation direction.
In this way, alternative expressions for the pressure changes in M 1 and M 2 are
obtained which are summarized in the tables below. Once again the equation for M 2
is multiplied by (−1) and summed with that for M 1 to eliminate the variable P v .
Mesh 1 (M 1 )
Corrected approach
+R 12 Q 2
12
+R 12 Q 12 |Q 12 |
−R 24 Q 2
24
−R 24 Q 24 |Q 24 |
−R 14 (Q e − Q 12 ) 2
−R 14 (Q e − Q 12 ) |Q e − Q 12 |
(−1) · Mesh 2 (M 2 )
Corrected approach
+R 23 (Q 12 + Q 24 ) 2
+R 23 (Q 12 + Q 24 ) |Q 12 + Q 24 |
+R 34 ((Q 24 − Q e ) + Q 12 ) 2
+R 34 (Q 24 − Q e + Q 12 ) |Q 24 − Q e + Q 12 |
+R 24 Q 2
24
+R 24 Q 24 |Q 24 |
Unlike the previous method, here, we end up with an equation effectively
containing two unknowns, the variable Q i and the absolute value of Q i . This, of
course, has no direct solution. To get over this difficulty, we use the Solver Solution
Search System from Excel for its simplicity. Entering the values from the problem
into the equation, we have found from the mesh analysis and imposing the condition
that Q 12 ≤ Q e , (that is Q 12 ≤ 150) at all times, that:
Mesh 1
Mesh 2
M 1 + M 2
f (solver):
−3216.83
3216.83
0.00
Q 12
68.80