Lagrange Multipliers 497
which gives
L
*
1
25
0.04 m 4 cm
The second derivative is given by
dQ
dL
LL
2
2
94 74
861
5
8
15
8
¤
¦
¥
³
µ
´
.
//
At L
L
L 00
56LT$.
These equations can be solved to yield the optimum as
L
*
0.04 m; $T
*
140; Q
*
77.05 W; L17.2
It can be shown that if the constraint is increased from 5.6 to 5.7, the heat transfer
rate Q becomes 78.77, i.e., an increase of 1.72. This change can also be obtained
from the sensitivity coefcient S
c
. Here, S
c
–L 17.2, which gives the change in Q
Q
L
Minimum
FIGURE 8.7 Variation of the heat transfer rate Q with dimension L of the power source
in Example 8.5.
498 Design and Optimization of Thermal Systems
for a change of 1.0 in the constraint. Therefore, for a change of 0.1, Q is expected
to increase by 1.72.
Example 8.6
For the solar energy system considered in Example 5.3, the cost U of the system is
given by the expression
´
l()
.
/
35
5833 3
290 100
208 Minimum
or
U
V
V
l
2041 67
29 1
208
.
./
Minimum
Therefore, U may be differentiated with respect to V and the derivative set equal to
zero to obtain the optimum. This leads to the equation
2041 67
29 1
1
208
22
.
(. / )
Lagrange Multipliers 499
8.5.3 I NEQUALITY CONSTRAINTS
Inequality constraints arise largely due to limitations on temperature, pressure,
heat input, and other quantities that relate to material strength, process require-
ments, environmental aspects, and space, equipment, and material availability.
For instance, the temperature T
o
of cooling water at the condenser outlet of a
power plant is constrained due to environmental regulations as T
o
T
amb
R,
where T
amb
is the ambient temperature and R is the regulated temperature differ-
ence. The outlet from a cooling tower has a similar constraint. Other common
constraints such as
T a T
max
, P a P
max
, TqT
min
,
m
m
value in order to satisfy the constraint. For instance, if the temperature and
pressure in an extruder are restricted by strength considerations, we cannot
use this information to set the conditions at certain locations because it is not
known a priori where the maxima occur. In such cases, the common approach
is to solve the problem without considering the inequality constraint and then
checking the solution obtained if the constraint is satised. If not, the design
variables obtained for the optimum are adjusted to satisfy the constraint. If even
this does not work, the solution obtained may be used to determine the locations
where the constraint is violated, set the values at these points at less than the
maximum or more than the minimum, and solve the problem again. With these
efforts, the inequality constraints are often satised. However, if even after all
these efforts the constraints are not satised, it is best to apply other optimiza-
tion methods.
500 Design and Optimization of Thermal Systems
8.5.4 S OME PRACTICAL CONSIDERATIONS
In the preceding discussion, we assumed that an optimum of the objective func-
tion exists in the design domain and methods for determining the location of
this optimum were obtained. However, many different situations may and often
do arise when dealing with practical thermal systems. Frequently, for uncon-
strained problems, several local maxima and minima are present in the domain,
which is dened by the ranges of the design variables, as sketched in Figure8.8.
These optima are determined by solving the system of algebraic equations
derived from the vector equation U 0. Since nonlinear equations generally
arise for thermal systems and processes, multiple solutions may be obtained,
indicating different local optima. Since interest obviously lies in the overall
or global maximum or minimum, it is necessary to consider each extremum in
order to ensure that the global optimum has been obtained. Multiple solutions
are also possible for constrained problems because of the generally nonlinear
nature of the equations. Again, each optimum point must be considered and the
objective function determined so that the desired best solution over the entire
1
and P
2
(see Figure5.38). Similarly, energy balance for mate-
rials undergoing heat treatment may yield a temperature, beyond the allowable
range, at which the system heat loss is minimized. In such cases, the maximum
or minimum allowable values of the design parameters that result in the larg-
est or smallest value of the objective function, as desired, are chosen for the
design.
8.5.5 C OMPUTATIONAL APPROACH
Analytical methods for deriving and solving the equations for the Lagrange mul-
tiplier method are generally applicable to a relatively small number of compo-
nents and simple expressions. A computational approach may be developed for
problems that are more complicated. One such scheme is based on the solution
of a system of nonlinear equations by the Newton-Raphson method, presented
in Chapter 4. The governing equations from the method of Lagrange multipliers
may be written as
FIGURE 8.9 Monotonically varying objective functions over given acceptable domains,
resulting in optimum at the boundaries of the domain.
Acceptable domain
U
x
Acceptable domain
U
A
A
x
502 Design and Optimization of Thermal Systems
Fxx x
U
Fxx x
U
m
m
nm
11
212 1
0$
##
L
LL(, , , , , , )
xx
G
x
G
x
G
x
Fxx
m
m
i
2
1
1
2
2
2
22
12
t
t
t
mm
m
i
nj n m j
G
x
Fxx x Gxx
t
t
0
12 1 12
(, , , , , , ) (,##LL ,, , )# x
n
0
(8.57)
where
i 1, 2, , n and j 1, 2, , m
Therefore, a system of n m equations is obtained, with the n independent vari-
ables and the m multipliers as the unknowns. These equations may be solved by start-
ing with guessed values of the unknowns and solving the following system of linear
equations for the changes in the unknowns, $x
i
m
n
1
1
1
2
1
12
$
%
$
L
L
11
1
1
2
1
12
t
t
t
t
t
t
t
t
t
t
¥
¥
¥
¥
³
µ
´
´
´
´
´
´
´
´
´
´
´
´
´
m
m
n
x
x
L
$
$
1
%
$$
´
´
´
´
´
¤
¦
¥
¥
¥
¥
¥
¥
¥
³
µ
´
´
´
´
´
´
´
i
, the rst derivative may be obtained from
t
t
f
x
fxx x x x fxx x
i
j
ijjni
(,,, ,,) (,,,
12 12
## #$
jjn
j
x
x
,,)#
$
(8.60)
where $x
j
is a chosen small increment in x
j
. Second derivatives will also be
needed because the functions F
i
in Equation (8.57) contain rst derivatives. The
jn
j
##
$
(8.61)
Other nite difference approximations can also be used, as discussed in Chapter 4.
Therefore, a numerical scheme may be developed to determine the optimum
using the method of Lagrange multipliers. The guessed values are entered and
the iteration process is carried out until the unknowns do not change signicantly
from one iteration to the next, as given by a chosen convergence criterion (see
Chapter 4). However, the process is quite involved because the rst and second
derivatives may have to be obtained numerically and a system of linear equations
is to be solved for each iteration. Such an approach is suitable for complicated
expressions and for a relatively large number of independent variables and con-
straints, generally in the range 5–10. For a still larger number of unknowns, the
problem becomes very complicated and time consuming, making it necessary to
seek alternative approaches.
8.6 SUMMARY
This chapter focuses on the calculus-based methods for optimization. These meth-
ods use the derivatives of the objective function U and the constraints to determine
the location where the objective function is a minimum or a maximum. For the
unconstrained problem, a stationary point is indicated by the partial derivatives of
the objective function U, with respect to the independent variables, going to zero.
The nature of the stationary point, whether it is a maximum, a minimum, or a
saddle point, is determined by obtaining the higher-order derivatives. For the con-
strained problem, the method of Lagrange multipliers is introduced and the system
of equations, whose solution yields the optimum, is derived. Derivatives are again
needed, making it a requirement for applying calculus methods that the objective
function and the constraints must be continuous and differentiable. In addition,
only equality constraints can be treated by this approach. The importance of this
REFERENCES
Beightler, C.S., Phillips, D., and Wilde, D.J. (1979) Foundations of Optimization , 2nd ed.,
Prentice-Hall, Englewood Cliffs, NJ.
Chong, E.K.P. and Zot, S.H. (2001) An Introduction to Optimization , 2nd ed.,
Wiley-Interscience, New York.
Dieter, G.E. (2000) Engineering Design , 3rd ed., McGraw-Hill, New York.
Fox, R.L. (1971) Optimization Methods for Engineering Design , Addison-Wesley,
Reading, MA.
Gebhart, B. (1971) Heat Transfer , 2nd ed., McGraw-Hill, New York.
Gerald, C.F. and Wheatley, P.O. (2003) Applied Numerical Analysis , 7th ed., Addison-
Wesley, Reading, MA.
Jaluria, Y. (1996) Computer Methods for Engineering , Taylor & Francis, Washington, D.C.
Kaplan, W. (2002) Advanced Calculus , 5th ed., Addison-Wesley, Reading, MA.
Keisler, H.J. (1986) Elementary Calculus , 2nd ed., PWS, Boston, MA.
Stoecker, W.F. (1989) Design of Thermal Systems , 3rd ed., McGraw-Hill, New York.
Lagrange Multipliers 505
PROBLEMS
8.1. The cost C involved in the transportation of hot water through a pipe-
line is given by
C
D
Dx
D
xD
¤
¦
¥
³
µ
Insulation
FIGURE P8.1
506 Design and Optimization of Thermal Systems
8.3. The cost C in a metal forming process is given in terms of the speed U
of the material as
C
KS
U
U
Ô
Ư
Ơ
à
Đ
â
ă
ă
ả
á
ã
ã
35
2173
3
18
16
mum or a maximum.
8.6. A rectangular duct of length L and height H is to be placed in a trian-
gular region of each side equal to 1.0 m, as shown in Figure P8.6, so
1m 1m
1m
L
H
FIGURE P8.6
Lagrange Multipliers 507
that the cross-sectional area of the duct is maximized. Formulate the
optimization problem as a constrained circumstance and determine
the optimal dimensions.
8.7. A rectangular box has a square base, with each side of length L,
and height H. The volume of the box is to be maximized, provided
the sum of the height and the four sides of the base does not exceed
100 cm, i.e., H 4L 100. Set up the optimization problem and cal-
culate the dimensions at which maximum volume is obtained.
8.8. Consider the convective heat transfer from a spherical reactor of
diameter D and temperature T
s
to a uid at temperature T
a
, with a
convective heat transfer coefcient h. Denoting ( T
s
– T
a
) as Q, h is
given by
h 2 0.55 Q
¤
¦
¥
³
µ
´
387
13
32
13
12
/
/
/
/
.
It is also given that the temperature T must not exceed 7.5 L
3/4
.
Calculate the dimension L that minimizes the heat loss, treating the
problem as an unconstrained one rst and then as a constrained one.
What information does the Langrange multiplier yield in the latter
case?
8.10. For the solar energy system considered in Example 8.6, study the
effect of varying the cost per unit surface area of the reactor, given
as 35 in the problem, and also of varying the cost per unit volume of
the storage tank, given as 208. Vary these quantities by o20% of the
given values in turn, keeping the other coefcient unchanged, and
508 Design and Optimization of Thermal Systems
where
m
1
and
m
2
are the ow rates through the two pipes. If the
total heat input is to be minimized, set up the optimization problem
for this system. Using Lagrange multipliers for a constrained prob-
lem, obtain the optimal values of the ow rates and the sensitivity
coefcient. What does it represent physically in this problem?
8.13. The mass ow rates in two pipes are denoted by
m
1
and
m
2
. The
heat inputs in these two circuits are correspondingly given as q
1
and
q
2
m
2
that optimize the total heat input,
q
1
q
2
, using the method of Lagrange multipliers. Also, obtain the
sensitivity coefcient.
8.14. The fuel consumption F of a vehicle is given in terms of two param-
eters x and y, which characterize the combustion process and the
drag as
F 10.5 x
1.5
6.2 y
0.7
with a constraint from conservation laws as
x
1.2
y
2
20
Lagrange Multipliers 509
Cast this problem as an unconstrained optimization problem and
solve it by the Lagrange multiplier method. Is it a maximum or a
minimum?
8.15. In a water ow system, the total ow rate Y is given in terms of two
variables x and y as
Y 8.5 x
2
preference, nances available, reputation of the manufacturer, system features,
etc. In a similar way, optimization of practical thermal systems may be based on
considering a number of feasible designs and choosing the best one, as guided by
the objective function.
This chapter discusses the use of search methods for the optimization of ther-
mal systems. The basic approaches employed and the different methods available
are presented. Since generating a feasible design is generally a time-consuming
process, it is necessary to minimize the number of designs needed to reach the
optimum. Therefore, efcient search methods that converge rapidly to the opti-
mum have been developed and are extensively used for thermal systems. The
efciency of the different methods is also considered, in terms of iterative steps
needed to reach the optimum.
Both constrained and unconstrained problems are considered, for single as
well as multiple independent variables. As discussed in Chapter 8, a constrained
problem may often be transformed into an unconstrained one by using substitution
and elimination. In addition, the constraints are often included in the calculation
of the objective function from modeling and simulation, making the optimization
problem an unconstrained one. Thus, unconstrained problems, which are often
much simpler to solve than the constrained ones, arise in a wide variety of prac-
tical systems and processes. A brief discussion of search methods is given in
this chapter, along with a few examples to illustrate their application to thermal
512 Design and Optimization of Thermal Systems
systems. For further details on these methods, textbooks on optimization, such
as Siddall (1982), Reklaitis et al. (1983), Vanderplaats (1984), Rao (1996), Arora
(2004), and Ravindran et al. (2006), may be consulted.
9.1.1 IMPORTANCE OF SEARCH METHODS
In many practical thermal systems, the design variables are not continuous func-
tions but assume nite values over their acceptable ranges. This is largely due to
the limited number of materials and components available for design. Finite num-
bers of components, such as pumps, blowers, fans, compressors, heat exchangers,
mum design in a region whose boundaries are dened by the ranges of the design
variables. In order to illustrate the different methods, relatively simple expres-
sions are employed here for which search methods are not necessary, and simpler
schemes such as the calculus methods can easily be employed. However, this is
Search Methods 513
only for illustration purposes and, in actual practice, each test run or simulation
would generally involve considerable time and effort. Some practical systems
are also considered to demonstrate the application of these methods to more
complex systems.
9.1.2 TYPES OF APPROACHES
There are several approaches that may be employed in search methods, depend-
ing on whether a constrained or an unconstrained problem is being considered
and whether the problem involves a single variable or multiple variables. These
approaches may be classied as follows.
Elimination Methods
In these methods, the domain in which the optimum lies is gradually reduced by
eliminating regions that are determined not to contain the optimum. We start with
the design domain dened by the acceptable ranges of the variables. This region
is known as the initial interval of uncertainty. Therefore, the region of uncer-
tainty in which the optimum lies is reduced until a desired interval is achieved.
Appropriate values of the design variables are chosen from this interval to obtain
the optimal design. For single-variable problems, the main search methods based
on elimination are
Exhaustive search
Dichotomous search
Fibonacci search
Golden section search
All these approaches have their own characteristics, advantages, and applicabil-
ity, as discussed later in detail. These methods can also be used for multivariable
problems by applying the approach to one variable at a time. This technique,
must be satised while searching for the optimum. This restricts the movement
toward the optimum. The constraints may also dene the acceptable design domain.
Two important schemes for optimizing constrained problems are
Penalty function method
Searching along a constraint
The former approach combines the objective function and the constraints into a
new function which is treated as unconstrained, but which allows the effect of
the constraints to be taken into account through a careful choice of weighting
factors. The latter approach can be combined with the methods mentioned earlier
for unconstrained optimization, particularly with the steepest ascent method. The
search is carried out along the constraints so that the choices are limited and the
optimal design satises these constraints. The procedure becomes quite involved
in all but very simple cases. Therefore, effort is often directed at converting a con-
strained problem to an unconstrained one or the penalty function method is used.
9.1.3 APPLICATION TO THERMAL SYSTEMS
As discussed in preceding chapters, each simulation or experimental run is gener-
ally very involved and time consuming for practical thermal systems. For instance,
the temperature T
b
of the barrel in the screw extrusion of plastics (see Figure 1.10b)
is an important variable. If the optimum temperature is sought in order to maxi-
mize the mass ow rate or minimize the cost, simulation of the system must be
carried out at different temperatures, over the acceptable range, to choose the best
value. However, each simulation involves solving the governing partial differential
Search Methods 515
equations for the ow and heat transfer of the plastic in the extruder as well as in
the die. The material melts as it moves in the screw channel and the viscosity of
the molten plastic varies with temperature and shear rate in the ow, the latter
characteristic known as the non-Newtonian behavior of the uid. Similarly, other
properties are temperature dependent. Other complexities such as the complicated
in the preceding discussion, the barrel temperature T
b
may be allowed to range
from room temperature to the charring temperature of the plastic, which is around
250nC for typical plastics and is the temperature at which these are damaged.
The single-variable optimization problem is of limited interest in thermal sys-
tems because several independent variables are generally important in practical
circumstances. However, there are two main reasons to study the single-variable
problem. First, there are systems whose performance is dominated by a single
variable, even though other variables affect its performance. Examples of such a
dominant single variable are heat rejected by a power plant, energy dissipated in
an electronic system, temperature setting in an air conditioning or heating system,
516 Design and Optimization of Thermal Systems
pressure or concentration in a chemical reactor, fuel ow rate in a furnace, surface
area in a heat exchanger, and speed of an automobile. In such cases, the opti-
mal design may be sought by varying only the single, dominant variable. Second,
many multivariable optimization problems are solved by alternately optimizing
with respect to each variable.
If U(x) is a continuous, differentiable function, such as the ones shown in
Figure 9.1, the maximum or the minimum could easily be found by setting the
derivative dU/dx 0. However, in search methods discrete runs are made at vari-
ous values of x to determine the location of the optimum or the interval in which it
lies, to the desired accuracy level. The objective function may be unimodal in the
given domain, i.e., it has a single minimum or maximum, as sketched in Figure 9.1,
or it may have several such local minima or maxima, as seen in Figure 9.2. Most of
FIGURE 9.1 Unimodal objective function distributions, showing a maximum and a
minimum.
U
x
Maximum
the region between the location where the smaller value of U(x) is obtained in two
runs and the nearest boundary is eliminated, as shown in Figure 9.3 in terms of
the results from three runs. In Figure 9.3(a), the region beyond C and that before
A are eliminated, thus reducing the domain in which the maximum lies to the
region between A and C. Similarly, in Figure 9.3(b), the region between the lower
domain boundary and point B is eliminated.
Consider a chemical manufacturing plant in which the temperature T
r
in the
reactor determines the output M by shifting the equilibrium of the reaction. If
the temperature can be varied over the range 300 to 600 K, the initial region of
uncertainty is 300 K. The maximum output in this range is to be determined.
If ve trial points or runs are chosen, i.e., n 5, the range is subdivided into six
FIGURE 9.3 Elimination of regions in the search for a maximum in U.
U
x
Eliminate
Eliminate
(a) (b)
U
x
A
C
B
A
C
B
Eliminate
518 Design and Optimization of Thermal Systems
intervals, each of width 50 K, as shown in Figure 9.4. The output is computed
(9.3)
For the uniform exhaustive search method, the reduction ratio is
R
n
1
2
(9.4)
Therefore, the number of experiments or trial runs n needed for obtaining a desired
interval of uncertainty may be determined from this equation. For instance, if, in
Output (M)
T
r
(K)
300
50
100
400 500 600
FIGURE 9.4 Uniform exhaustive search for the maximum in the output M in a chemical
reactor, with the temperature T
r
as the independent variable.
Search Methods 519
the preceding example, the region containing the optimum is to be reduced to
30 K, then the reduction ratio is 10 and the number n of trial runs needed to
accomplish this is 19.
The exhaustive search method is not a very efcient strategy to determine the
optimum because it covers the entire domain uniformly. However, it does reveal
the general characteristics of the objective function being optimized, particularly
whether it is unimodal, whether there is indeed an optimum, and whether it is a
Here, the pairs A, a and B, b are located at equal distance on either side of the
chosen values of 400 and 500 K, with a difference of o5 K from these. The sepa-
ration E must be larger than the error in xing the value of the variable in order to
obtain accurate and repeatable results.
520 Design and Optimization of Thermal Systems
For n runs or simulations, the initial range L
o
is divided into (n/2) 1 sub-
intervals, neglecting the region between a single pair. Since the nal interval
of uncertainty L
f
has the width of a single subdivision, the reduction ratio R is
obtained, neglecting the separation E, as
R
n
2
1
(9.5)
Therefore, the initial interval of uncertainty is reduced to one-third, or
400 T
r
500 K, after four runs. With exhaustive search, 40% of the domain is
left after four runs, as seen from Equation (9.4). Therefore, the uniform dichoto-
mous search is slightly faster in convergence than the uniform exhaustive search.
However, the sequential dichotomous search, discussed next, is a considerable
improvement over both of these.
Sequential Dichotomous Search
As before, this method uses pairs of experiments or simulations to ascertain
whether the function is increasing or decreasing and thus reduce the interval con-
a
and M
A
are the
values of the objective function at these two points, the region to the left is elimi-
nated, and the new interval of uncertainty is 450 T
r
600 K if Eis neglected.
The next pair, B and b, is then placed at the middle of this domain, i.e., at T
r
525 K, as shown in Figure 9.6. Again, by inspection, since M
B
M
b
, the region to
the right of the pair is eliminated, leaving the interval 450 T
r
525. Therefore,
the interval of uncertainty is reduced to 25%, or one-fourth, of its initial value.
With each pair, the region of uncertainty is halved. Therefore, neglecting the
separation E, the interval is halved n/2 times, where n is the total number of runs
and is an even number. Therefore, the reduction ratio is obtained as R 2
n/2
. This
implies that an even number of runs may be chosen a priori to reduce the region
of uncertainty to obtain the desired accuracy in the selection of the independent
variable for optimal design.
9.2.3 FIBONACCI SEARCH
The Fibonacci search is a very efcient technique to narrow the domain in which
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