Design and Optimization of Thermal Systems Episode 3 Part 1 pot

This cost is expressed in terms of the mass flow rate m of the material as FIGURE 8.1 Sketches showing a maximum, a minimum, and an inflection point in a function yx plotted against th

We are all quite familiar, from courses in mathematics, with the determination of the maximum or minimum of a function by the use of calculus If the function is continuous and differentiable, its derivative becomes zero at the extremum For a

function y(x), this condition is written as

dy

where x is the independent variable The basis for this property may be explained

in terms of the extrema shown in Figure 8.1 As the maximum at point A is approached, the value of the function y(x) increases and just beyond this point,

it decreases, resulting in zero gradient at A Similarly, the value of the function decreases up to the minimum at point B and increases beyond B, giving a zero slope at B In order to determine whether the point is a maximum or a minimum,

the second derivative is calculated Since the slope goes from positive to negative, through zero, at the maximum, the second derivative is negative Similarly, the slope increases at a minimum and, thus, the second derivative is positive These conditions may be written as (Keisler, 1986)

These conditions apply for nonlinear functions y(x) and, therefore, calculus

meth-ods are useful for thermal systems, which are generally governed by nonlinear expressions However, both the function and its derivative must be continuous for the preceding analysis to apply

Thus, by setting the gradient equal to zero, the locations of the extrema may be obtained and the second derivative may then be used to determine the nature of each extremum There are cases where both the first and the sec-ond derivatives are zero This indicates an inflection point, as sketched inFigure 8.1(c), a saddle point, or a flat curve, as in a ridge or valley It must be noted that the conditions just mentioned indicate only a local extremum There may be several such local extrema in the given domain Since our interest lies in the overall maximum or minimum in the entire domain for optimizing the system, we would

seek the global extremum, which is usually unique and represents the largest or smallest value of the objective function The following simple example illustrates the use of the preceding procedure for optimization.

Example 8.1

Apply the calculus-based optimization technique just given to the minimization of

cost C for hot rolling a given amount of metal This cost is expressed in terms of the

mass flow rate m of the material as

FIGURE 8.1 Sketches showing a maximum, a minimum, and an inflection point in a

function y(x) plotted against the independent variable x.

which is positive because the flow rate mis positive This implies that the

optimi-zation technique has yielded a minimum of the objective function C, as desired

Therefore, minimum cost is obtained at m  1.692, and the corresponding value

of C is 11.962.

The preceding discussion and the simple example serve to illustrate the use of calculus for optimization of an unconstrained problem with a single independent variable However, such simple problems are rarely encountered when dealing with the optimization of practical thermal systems Usually, several independent variables are involved and constraints may have to be satisfied This consider-ably complicates the application of calculus to extract the optimal solution In addition, the use of calculus methods requires that any constraints in the problem must be equality constraints This limitation is often circumvented by convert-ing inequality constraints into equality ones, as outlined in Chapter 7 In many practical circumstances, the objective function is not readily available in the form

of continuous and differentiable functions, such as the one given in Example 8.1 However, curve fitting of numerical and experimental data may be used in some cases to yield continuous expressions that characterize the given system and that can then be used to obtain the optimum

Calculus methods, whenever applicable, provide a fast and convenient method

to determine the optimum They also indicate the basic considerations in zation and the characteristics of the problem under consideration In addition, some of the ideas and procedures used for these methods are employed in other techniques Therefore, it is important to understand this optimization method and the basic concepts introduced by this approach This chapter presents the Lagrange multiplier method, which is based on the differentiation of the objec-tive function and the constraints The physical interpretation of this approach

optimi-is brought out and the method optimi-is applied to both constrained and unconstrained optimization The sensitivity of the optimum to changes in the constraints is dis-cussed Finally, the application of this method to thermal systems is considered

8.2 THE LAGRANGE MULTIPLIER METHOD

This is the most important and useful method for optimization based on calculus

It can be used to optimize functions that depend on a number of independent variables and when functional constraints are involved As such, it can be applied

to a wide range of practical circumstances provided the objective function and the constraints can be expressed as continuous and differentiable functions In addi-tion, only equality constraints can be considered in the optimization process

8.2.1 B ASIC A PPROACH

The mathematical statement of the optimization problem was given in the ing chapter as

preced-U(x , x , x , , x)l Optimum (8.4)

subject to the constraints

The method of Lagrange multipliers basically converts the preceding problem

of finding the minimum or maximum into the solution of a system of algebraic equations, thus providing a convenient scheme to determine the optimum The

objective function and the constraints are combined into a new function Y, known

as the Lagrange expression and defined as

Y(x1, x2, , x n) U(x1, x2, , x n)1G1(x1, x2, , x n) L2G2(x1, x2, , x n)

z Lm G m (x1, x2, , x n) (8.6)where the L’s are unknown parameters, known as Lagrange multipliers Then, according to this method, the optimum occurs at the solution of the system of equations formed by the following equations:

Y x

G x

G x

G x U

x

1

1 11

2 2

1 1

0t

G x

G x

G x

U x

G x

G1(x1, x2, x3, , x n) 0

G2(x1, x2, x3, , x n) 0 (8.8)

G m (x1, x2, x3, , x n) 0

If the objective function U and the constraints G i are continuous and

differen-tiable, a system of algebraic equations is obtained Since there are m equations for the constraints and n additional equations are derived from the Lagrange expres- sion, a total of m n simultaneous equations are obtained The unknowns are the

m multipliers, corresponding to the m constraints, and the n independent

vari-ables Therefore, this system may be solved by the methods outlined in Chapter 4

to obtain the values of the independent variables, which define the location of the optimum, as well as the multipliers Analytical methods for solving a system

of algebraic equations may be employed if linear equations are obtained and/or when the number of equations is small, typically up to around five For nonlinear equations and for larger sets, numerical methods are generally more appropriate The optimum value of the objective function is then determined by substituting

the values obtained for the independent variables into the expression for U The optimum is often represented by asterisks, i.e., x1, x 2 , , x n , and U*

8.2.2 P HYSICAL I NTERPRETATION

In order to understand the physical reasoning behind the method of Lagrange

mul-tipliers, let us consider a problem with only two independent variables x and y and a single constraint G(x, y)  0 Then the optimum is obtained by solving the equations

t

t 

t

t t

G x U y

G y

G x y

LL

0

00( , )

where i and j are unit vectors in the x and y directions, respectively.

Therefore, 

two components in these directions For example, if the temperature T in a region

is given as a function of x and y, the rate of change of T in the two coordinate

directions is the components of the gradient vector 

effectively in heat conduction to represent the heat flux vector q, which is given

rep-ent vector, as well as a unit vector n in its direction, may be calculated as

Let us now consider a F  constant curve in the x-y plane, as shown in

Figure 8.2 for three values c1, c2, and c3 of this constant Then, from the chain rule in calculus,

FF

/

Therefore, the tangential vector T shown in Figure 8.2, may be obtained by

using a differential element dT is given by this relationship between dx and dy.

FF/

The unit vector t along the tangential direction may be obtained, as done

previ-ously for the gradient vector, by dividing the vector by its magnitude Thus,

///

where c and d represent the respective components given in the preceding

equations The relationship given by Equation (8.17) applies for vectors that are normal to each other This is shown graphically in Figure 8.3(a) Mathematically,

if a dot product of two vectors that are perpendicular to each other is taken, the

result should be zero Applying the dot product to n and t, we get

since i and j are independent of each other This confirms that the two tors t and n are perpendicular Therefore, the gradient vector 

vec-the constant F curve, as shown in Figure 8.3(b) This information is useful in

FIGURE 8.2 Contours of constant F shown on an x-y plane for different values of the

constant Also shown is the tangent vector T, which is tangential to one such contour.

Tangent vector

understanding the basic characteristics of the Lagrange multiplier method and for developing other optimization techniques, as seen in later chapters.

If three independent variables are considered, a surface is obtained for a stant value of F Then, the gradient vector considerations apply for a larger number of independent variables The gradient

x1 1 x x x n

2 2 3 3

where i1, i2, , i n are unit vectors in the n directions representing the n pendent variables x1, x2, , x n, respectively Therefore, these unit vectors are independent of each other Though it is difficult to visualize the gradient vector for more than three independent variables, the mathematical treatment of the problem is the same as that given previously for two independent variables

inde-Again, the n and t unit vectors may be determined and their dot product taken

contours or surfaces Because of this property, the gradient vector represents the direction in which the dependent variable F changes at the fastest rate, this rate being given by the magnitude of the gradient In addition, the direction in which F increases is the same as the direction of the vector ties are useful in many optimization strategies, particularly in gradient-based search methods

Lagrange Multiplier Method for Unconstrained Optimization

Let us first consider the unconstrained problem for two independent variables

x and y Then the Lagrange multiplier method yields the location of the optimum

FIGURE 8.3 (a) Unit vectors t and n are perpendicular to each other; (b) gradient vector

F is normal to the F = constant contour.

y

φ(x, y) = constant d

c

–d

c

n t

as the solution to the equation

U y

which may be solved to obtain x and y at the optimum, denoted as x* and y* The

optimal value U* is then calculated from the expression for U The number of

equations obtained is equal to the number of independent variables and the mum may be determined by solving these equations

opti-Lagrange Multiplier Method for Constrained Optimization

The optimum for a problem with a single constraint is obtained by solving the equations



 = 0

(b)

The gradient vector U is normal to the constant U contours, whereas G is

a vector normal to the constant G contours The Lagrange multiplier L is

sim-ply a constant Therefore, this equation implies that the two gradient vectors are aligned, i.e., they are both in the same straight line The magnitudes could be dif-ferent and L can be adjusted to ensure that Equation (8.22) is satisfied However,

if the two vectors are not in the same line, the sum cannot be zero unless both vectors are zero This result is shown graphically in Figure 8.5 for a minimum

in U As one moves along the constraint, given by G 0, in order to ensure that the constraint is satisfied, the gradient G varies in direction The point where

it becomes collinear with U is the optimum At this point, the two curves are

tangential and thus yield the minimum value of U while satisfying the constraint Constant U curves below the constraint curve do not satisfy the constraint and those above it give values of U larger than the optimum at the locations where they intersect with the constraint curve Clearly, values of U smaller than that at

the optimum shown in the figure could be obtained if there were no constraint, in which case the governing equations would be obtained from Equation (8.20)

As an example, consider an objective function U of the form

FIGURE 8.5 Physical interpretation of the method of Lagrange multipliers for two

inde-pendent variables and a single constraint.

G = 0

4 3 2 1

U = 5

Here E, the coefficients A and B, and the exponents a, b, c, and d are assumed

to be known constants Such expressions are frequently encountered in thermal

systems For example, U may be the overall cost and x and y the pump needed

and pipe diameter, respectively, in a water flow system The pressure decreases

as the diameter increases, resulting in lower cost for the pump, and the cost for the pipe increases This gives rise to a relationship such as Equation (8.24)

Thus, contours of constant U may be drawn along with the constraint curve on

an x

the location where the constant U contour becomes tangential to the constraint

curve, thus aligning the U and G vectors For the simple case when all the

constants in these expressions are unity, i.e., U

the constant U contours are straight lines and the constraint curve is given by

x  1/y, as sketched in Figure 8.6(b) The optimum is at x* 1.0 and y* 1.0, and

the optimum value U* is 2.0 for this case

Even though only two independent variables are considered here for ease of visualization and physical understanding, the basic ideas can easily be extended

to a larger number of variables The system of equations to be solved to obtain the

optimum is given by the n scalar equations derived from the vector equation

and the m equality constraints

G i  0, for i  1, 2, 3, , m (8.27)

FIGURE 8.6 (a) Optimization of the simple constrained problem given by

Equa-tion (8.23) to EquaEqua-tion (8.25), and (b) the results when all the constants in the given expressions are unity.

3 2

1

2.0 1.0 0.0

U = x + y

= 4

U*= 2

G = xy – 1 = 0

These equations are solved analytically or numerically to yield the multipliers and the values of the independent variables, from which the optimum value of the objective function is determined.

Proof of the Method

The proof of the method of Lagrange multipliers is available in most books on optimization However, the mathematical analysis becomes involved as the num-ber of independent variables and the number of constraints increase (Stoecker, 1989) Let us again consider the problem with two independent variables and

a single constraint for simplicity At the desired optimum point, the constraint must also be satisfied If we now deviate from this point, while ensuring that the

constraint continues to be satisfied, the change in G should be zero, i.e., from the

¤

¦¥

³µ´ 

tt

¤

¦¥

³

where dx and dy are the changes in x and y Therefore, these changes are related

by the following expression, if the condition of G 0 is to be preserved:

¤

¦¥

³µ´ 

tt

¤

¦¥

³µ´

³³µ

t

t 

tt

where L is the Lagrange multiplier for this one-constraint problem In order for

the starting point to be the optimum, there should be no change in U for a ential movement from the optimum while satisfying the constraint G 0, i.e., this