Environmental Engineering Reference
In-Depth Information
In general the number of variables, n, will be greater than the number of equations, m,
and the difference between (n-m) is commonly denoted as the number of degrees of freedom
of the optimization problem. Any optimization problem can be represented in the above form.
If we want to maximize a function this is equivalent to minimizing the negative of that
function.
Now for the heating and cooling utilities minimization problem, let us go back to our
small problem solved algebraically before using pinch technology and use FCP for cold
streams of 19 and 2 kW/ºC resepctively. The new problem can be easily solved using
mathematical programming model. We write our objective function not only including
heating and cooling utiltities loads but also including heating and colling utilitie costs in
dollar value; formulate our model/constraints using the cascade approach; and then solving
the optimization problem using any commercial software.
•
Objective function
10
−
6
Q
min
heating
10
−
6
Q
min
Minimize (5*
+ 9*
)*3600*8000
cooling
•
Define the loads of heating and cooling utilities in each temperature interval and
the surplus from each interval as we did before in the algebraic method through
the development of temperature interval diagram, tables of exchangeable loads
and un-balanced thermal cascade diagram.
The model formulation is a heat balance around each temperature interval in the graph
below as follows:
min
heating
Q
0.0
760
1
r1
1300
2470
2
r2
100
210
3
r3
750
4
1050
r4
100
380
5
r5
50
0.0
6
min
cooling
Q
Figure 13. Thermal Cascade diagram for LP model
min
heatin
Q
+ 0.0-760 = r1
r1+1300-2470 = r2
