154
Chapter 3
Z 2
4xyz +/~1 ~ = 0
(3.17)
Adding Eqs. (3.15)-(3.17) and noting the last term is equal
//x 2 y2 ~-~ 7 .5 22)
12xyz + 2~ k~ + + = 0
(3.18)
Substituting 21 separately into Eqs. (3.12)-(3.14)
2a
2b
2c
2x = q-~ 2y = -l-v~
2z = :1:~
(3.19)
The regional constraint requires that only positive values are used.
V. OPTIMIZATION WITH NUMERICAL METHODS
Many times a problem [3.7,3.23] becomes so complex computer numerical
iterations are required for a solution.
EXAMPLE 3.4. A hot-water pipe line [3.7,3.23] is to be designed to
carry a large quantity of hot water from the heater to the point of use.
The cost in dollars per length consists of four items. In this case, only positive values are desired.
(a) Cost of pumping the water from pipe pressure losses
1
100
Cp = Kp D5 - D5
(3.20)
(b) Cost of heat lost from pipe from the heat transfer through the
wall
Ka
1
Ch = ln[(D + 2x)/D] -- ln[(D ÷ 2x)/D]
(3.21)
(c) Cost of pipe
Cpipe
= K3D = 0.50D
(3.22)
(d) Cost of insulation
CiK4x = 1.0 x
(3.23)
When the costs are summed
C= Cp .-{- Ch .~- Cpipe -{- C
i
The optimal solution [3.7] yields
(3.24)
D = 1.86 inches and x = 1.37 inches with C = 4.56 dollars per length