Engineering Optimization: Theory and Practice, Fourth Edition

54 Introduction to Optimization

The air gap is to be less thank 1

√

x 2 + 7 .5 wherek 1 is a constant. The temperature of

the external surface of the motor cannot exceed

T above the ambient temperature.

Assuming that the heat can be dissipated only by radiation, formulate the problem for

maximizing the power of the motor [1.59].Hints:

1.The heat generated due to current flow is given byk 2 x 1 x 2 −^1 x− 41 x^25 , wherek 2 is a

constant. The heat radiated from the external surface for a temperature difference of



Tis given byk 3 x 1 x 2 

T, wherek 3 is a constant.

2.The expression for power is given byk 4 NBx 1 x 3 x 5 , wherek 4 is a constant,Nis the

rotational speed of the rotor, andBis the average flux density in the air gap.

3.The units of the various quantities are as follows. Lengths: centimeter, heat generated,

heat dissipated; power: watt; temperature:◦C; rotational speed: rpm; flux density:

gauss.

1.18 A gas pipeline is to be laid between two citiesAandE, making it pass through one

of the four locations in each of the intermediate townsB, C, andD(Fig. 1.23). The

associated costs are indicated in the following tables.

Costs forAtoBandDtoE

Stationi

1 2 3 4

FromAto pointiofB 30 35 25 40

From pointiofDtoE 50 40 35 25

Costs forBtoCandCtoD

To:

From: 1 2 3 4

1 22 18 24 18

2 35 25 15 21

3 24 20 26 20

4 22 21 23 22

Figure 1.23 Possible paths of the pipeline betweenAandE.