**UFBA**

# Applied Numerical Methods with MATLAB

(Parte **5** de 16)

Aside from Newton's second law. there are other major organizing principles in science and engineering. Among the most important of these are the conserv,ation lan:s. Although they form the basis for a variety of complicated and powerful mathematical models, the great conservation laws of science and engineering are conceptually easy to understand. They all boil down to

Change : increases - decreases(1.r4)

This is precisely the fbrmat that we empioyed when using Newton's law to develop a force balance for the bungee jumper tEq. ( 1.8)1. Although simple, Eq. (1.14) embodies one of the most fundarnental ways in which conservation laws are used in engineering and science-that is. to predict changes with respect to time. We will give it a special name-the time-variable (or transient) computation.

Aside from predicting changes, another way in which conservation laws are applied is fbr cases where change is nonexistent. If change is zero, Eq. (I.14) becomes

Change : 0 : increases - decreases or Increases : decreases(1.r5)

Thns, ifno change occurs, the increases and decreases nrust be in balance. This case, which is also given a special narne-the stea(ly-state calculation-has many applications in engineering and science. For example, fbr steady-state incompressible fluid flow in pipes, the flow into a junction musl be balanced by flow going out. as in

Flow in : flow out

For the junction in Fig. I .5, the balance can be used to compute that the flow out of the fourth pipe must be 60. For the bungee jumper, the steady-state condition would correspond to the case where the net lbrce was zero or [Eq. (1.8) with du ldt : 0l l1l$ : 6411-(1.r6)

Thus. at steady state, the downward and upward fbrces are in balance and Eq. ( 1. | 6) can be solved for the terminal velocity

Although Eqs. (1.14) and (1.15) might appeartrivially simple, they embody the two fundamental ways that conservation laws are employed in engineering and science. As such, they will tbnn an important part of our et-forts in subsequent chapters to illustrate the connection between numerical methods and engineering and science.

I.3 NUMERICAL METHODS COVERED IN THIS BOOKr3

Pipe 2 Flow in = 80

Pipe 1

Flow in : 100

Pipe 4 Flow out =

FIGURE I.5 A f ow bo once for steody incompressibe f uid flotv ot the junclion of pipes

Table l. I summarizes some models and associated conservation laws that figure promrnently in engineering. Many chemical engineering problems involve mass balances for reactors. The mass balance is derived from the conservation of mass. It specifies that the

change of mass of a chemical in the reactor depends on the amount of mass flowing in minus the n.rass flowing out.

Civil and mechanical engineers often focus on models developed from the conservation of momentum. Forcivil engineering, force balances are utilized to analyze structures such as the simple truss in Table 1.1. The same principles are employed for the mechanical engineering case studies to analyze the transient up-and-down motion or vibrations of an automobile.

Finally. electrical engineering studies en-rploy both current and energy balances to model electric circuits. The current balance, which results from the conservation of charge, is simi- lar in spirit to the flow balance depicted in Fig. 1.5. Just as flow mnstbalance at the junction of pipes, electric current must balance at the junction of electric wires. The energy balance specifies that the clranges of voltage around any loop of the circuit must add up to zero.

We should note that there are many otherbranches of engineering beyond chemical, civi,, electrical, and mechanical. Many of these ale related to the Big Four. For exalnple, chemical engineering skills are used extensively in areas such as environmental, petroleum, and biornedical engineering. Sirnilarly, aerospace engineering has much in cornmon with mechanical engineering. We will endeavor to include examples from these areas in the coming pages.

I.3 NUMERICAT METHODS COVERED IN THIS BOOK

We chose Euler's method for this introductory chapter because it is typical of many other classes of numerical methods. In essence, most consist of recasting mathematical operations into the simple kind of algebraic and logical operations compatible with digital compllters. Figure 1.6 summarizes the major areas covered in this text.

Pipe 3 Flow out : 120 l4MATHEMATICAT MODELING, NUMERICAT METHODS, AND PROBLEM SOLVING

TABTE l.l Devices ond types of bolonces fhot ore commonly used in ihe four moior oreos of engineering. For eoch cose, lhe conservotion low on which the bolonce is bosed is specified.

Field Device OrganizingPrinciple MathematicalExpression

Chemical engineering

Civil engi neeri ng

Mechanical engineering

Electrical engineering ffi?7fu. ,1m77, ilH3

Conservation of mass

Conservation of momentum

Conservation of momentum

Conservation of charge

Conservation of energy

Force balance:

Current balance: +i,

For each node I current (i) = 0

Voltage balance: a{A&-l ,,R, J -- f--2r*zY

L--\A7\--J i:R:

Around each loop I emf's - I voltage drops for resistors

>6->a:0

Mass balance: ffi inort ff_--* ourpur

Over a unit of time period Amass:inputs-outputs

+Fv

-Fn * O+ +FH

IV -at/

At each node

I horizontal forces (FH) = o I vertical forces (I'u) : 0

Force balance: I Upward force

(Parte **5** de 16)