4.10 Complex Networks
151
Application of the Model to Ventilation Networks
The application of the system to the solution of ventilation networks has similarities
with the previous cases explored, starting as it does from the principles of conservation of mass (Eq. 4.23) and energy (Eq. 4.24). In this case, as before, the expression
for energy conservation must be written using absolute values (see Eq. 4.25), leading
to Eq. 4.32:
F(Q i ) = R i |Q i |Q i
(4.32)
With regards to application of the law of conservation of mass to the nodes, it
must be the case that the derivative (Eq. 4.33) must be a constant, k:
∂ F
∂ Q i
= ±k
(4.33)
This derivative is necessary to calculate the Jacobian, and therefore, the term |Q|
Q in Eq. 4.32 requires further analysis.
Thus, the partial derivative of F(Q i ) with respect to a generic flow rate (Q i ) will
be:
∂ F
∂ Q i
= R i |Q i | + R i Q i |Q i |
1−1 ∂|Q i |
∂ Q i
Since we are working with absolute values there are two options: (a) that Q i is
positive and (b) that Q i is negative.
(a) If Q i > 0, then |Q i | = Q i , and the partial derivative
∂|Q i |
∂ Q i
= +1
And the equation remains:
∂ F
∂ Q i
= R i |Q i | + R i Q i |Q i |
1−1
· 1
From which we obtain Eq. 4.34:
∂ F
∂ Q i
= 2R i |Q i |
(4.34)
(b) If, on the other hand, Q i < 0, then |Q i | = −Q i , and the partial derivative
∂|Q i |
∂ Q i
= −1
Then we have:
∂ F
∂ Q i
= R i |Q i | + R i Q i |Q i |
1−1
(−1)
Which, because Q i can be simplified and then multiplied by (−1) to give Eq. 4.34.