Engineering Optimization: Theory and Practice, Fourth Edition

676 Optimal Control and Optimality Criteria Methods

To formulate the problem, we first write the heat balance equation for an elemental

length,dx, of the fin:

heat inflow by conduction = heat outflow by conduction and convection

that is,

(

−kA

dt

dx

)

x

=

(

−kA

dt

dx

)

x +dx

+ hS (t− t∞) (E 2 )

wherekis the thermal conductivity,Athe cross-sectional area of the fin= 2 y(x)

per unit width of the fin,hthe heat transfer coefficient,Sthe surface area of the fin

element= 2

√

1 +(y′)^2 dx er unit width, and 2p y(x)the depth of the fin at any section

x. By writing

(

−kA

dt

dx

)

x +dx

=

(

−kA

dt

dx

)

x

+

d

dx

(

−kA

dt

dx

)

dx (E 3 )

and noting thatt∞= , we can simplify Eq. (E 0 2 ) as

d

dx

(

ky

dt

dx

)

=ht

√

1 +(y′)^2 (E 4 )

Assuming thaty′≪ 1 for simplicity, this equation can be written as

k

d

dx

(

y

dt

dx

)

=ht (E 5 )

The amount of heat dissipated from the fin to the surroundings per unit time is

given by

H= 2

∫L

0

ht dx (E 6 )

byassuming that the heat flow from the free end of the fin is zero. Since the mass of

the fin is specified asm, we have

2

∫ L

0

ρy dx−m= 0 (E 7 )

whereρis the density of fin.

Now the problem can be stated as follows: Findt (x)that maximizes the integral

in Eq. (E 6 ) subject to the constraint equation (E 7 ). Since y(x)in Eq. (E 7 ) is also not

known, it can be expressed in terms oft (x)using the heat balance equation (E 5 ). By

integrating Eq. (E 5 ) between the limitsxandL, we obtain

−ky(x)

dt

dx

(x) = h

∫L

x

t (x) dx (E 8 )

by assuming the heat flow from the free end to be zero. Equation (E 8 ) gives

y(x)= −

h

k

1

dt/dx

∫L

x

t (x) dx (E 9 )