138
4 Mine Ventilation Networks
such functions to oscillate about the x-axis. Functions of the type Z = X
2 Y
2 ,
although more stable, are, however, very sensitive to the initial value chosen.
5. The function must be balanced, or compensated, with regards to what each of
the terms represents. Thus, it would not be correct to look for a solution to our
problem using a function of the type Z = X ln(Y ), since this would give one of
the meshes more weight than the other.
6. The chosen function should avoid errors as far as possible. Thus, when considering functions containing sums it should be remembered that a positive value
for X may occur together with a negative value for Y giving a solution of zero (or
close to zero). Such as solution would have no physical meaning. This situation
is avoided by squaring X and Y which always gives positive values.
Exercise 4.14 Taking the network shown in Exercise 4.13, and removing the fan from
branch 2–4, calculate the airflow rates in each branch using Newton’s method.
14
Using Kirchhoff’s first law:
Node
Equation
Substituted variable
N 1
Q a + Q 14 = Q 12
Q 14 = Q 12 − Q a
N 3
Q 23 = Q s (or Q a ) + Q 34
Q 34 = Q 23 − Q a
N 2
Q 12 = Q 24 + Q 23
Q 24 = Q 12 − Q 23
Equations in the nodes with the initial condition:
|Q a | = |Q s |
Using Kirchhoff’s second law:
Mesh 1: R 12 |Q 12 |Q 12 + R 24 |Q 24 | Q 24 + R 14 |Q 14 |Q 14 = 0
Mesh 2: R 23 |Q 23 | Q 23 + R 34 |Q 34 | Q 34 − R 24 |Q 24 |Q 24 = 0
Substituting variables:
Mesh 1:
R 12 |Q 12 | Q 12 + R 24 |Q 12 − Q 23 | Q 12 − Q 23 + R 14 |Q 12 − Q a | (Q 12 −
Q a ) = 0
Mesh 2:
R 23 |Q 23 | Q 23 + R 34 |Q 23 − Q a |(Q 23 − Q a ) − R 24 |Q 12 − Q 23 | (Q 12 −
Q 23 ) = 0
Note that to avoid losing negative signs during the squaring process we have used absolute
values
Generating the function: Z = X 2 + Y 2 , wherein:
X = R 12 |Q 12 | Q 12 + R 24 |Q 12 − Q 23 |(Q 12 − Q 23 ) + R 14 |Q 12 − Q a | (Q 12 − Q a ) = 0
Y = R 23 |Q 23 | Q 23 + R 34 |Q 23 − Q a |(Q 23 − Q a ) − R 24 |Q 12 − Q 23 | (Q 12 − Q 23 ) = 0
The solution of this function can then be found through a numerical optimization method,
where Q 12 and Q 23 are the variables to be optimized
Variable
Value
Solution
Branch
Quantity (m 3 s −1 )
Q 12 =
80.70
Q a = Q s
150.00
(continued)
14 By modifying the flow rate Q 24 , the system becomes unsolvable by traditional methods. Usually,
a system is generated in x, y, x · y, x 2 , y 2 , with roots from the above variables, which requires the
use of numerical calculation methods
4 Mine Ventilation Networks
such functions to oscillate about the x-axis. Functions of the type Z = X
2 Y
2 ,
although more stable, are, however, very sensitive to the initial value chosen.
5. The function must be balanced, or compensated, with regards to what each of
the terms represents. Thus, it would not be correct to look for a solution to our
problem using a function of the type Z = X ln(Y ), since this would give one of
the meshes more weight than the other.
6. The chosen function should avoid errors as far as possible. Thus, when considering functions containing sums it should be remembered that a positive value
for X may occur together with a negative value for Y giving a solution of zero (or
close to zero). Such as solution would have no physical meaning. This situation
is avoided by squaring X and Y which always gives positive values.
Exercise 4.14 Taking the network shown in Exercise 4.13, and removing the fan from
branch 2–4, calculate the airflow rates in each branch using Newton’s method.
14
Using Kirchhoff’s first law:
Node
Equation
Substituted variable
N 1
Q a + Q 14 = Q 12
Q 14 = Q 12 − Q a
N 3
Q 23 = Q s (or Q a ) + Q 34
Q 34 = Q 23 − Q a
N 2
Q 12 = Q 24 + Q 23
Q 24 = Q 12 − Q 23
Equations in the nodes with the initial condition:
|Q a | = |Q s |
Using Kirchhoff’s second law:
Mesh 1: R 12 |Q 12 |Q 12 + R 24 |Q 24 | Q 24 + R 14 |Q 14 |Q 14 = 0
Mesh 2: R 23 |Q 23 | Q 23 + R 34 |Q 34 | Q 34 − R 24 |Q 24 |Q 24 = 0
Substituting variables:
Mesh 1:
R 12 |Q 12 | Q 12 + R 24 |Q 12 − Q 23 | Q 12 − Q 23 + R 14 |Q 12 − Q a | (Q 12 −
Q a ) = 0
Mesh 2:
R 23 |Q 23 | Q 23 + R 34 |Q 23 − Q a |(Q 23 − Q a ) − R 24 |Q 12 − Q 23 | (Q 12 −
Q 23 ) = 0
Note that to avoid losing negative signs during the squaring process we have used absolute
values
Generating the function: Z = X 2 + Y 2 , wherein:
X = R 12 |Q 12 | Q 12 + R 24 |Q 12 − Q 23 |(Q 12 − Q 23 ) + R 14 |Q 12 − Q a | (Q 12 − Q a ) = 0
Y = R 23 |Q 23 | Q 23 + R 34 |Q 23 − Q a |(Q 23 − Q a ) − R 24 |Q 12 − Q 23 | (Q 12 − Q 23 ) = 0
The solution of this function can then be found through a numerical optimization method,
where Q 12 and Q 23 are the variables to be optimized
Variable
Value
Solution
Branch
Quantity (m 3 s −1 )
Q 12 =
80.70
Q a = Q s
150.00
(continued)
14 By modifying the flow rate Q 24 , the system becomes unsolvable by traditional methods. Usually,
a system is generated in x, y, x · y, x 2 , y 2 , with roots from the above variables, which requires the
use of numerical calculation methods