YOURSELF

2.5

Numerical

Calculation

and

Estimation 15 Finally,aruleof

thumb

forroundingoff

numbers

inwhichthedigitto

be dropped

isa 5 is

alwaysto

make

thelast digitof therounded-off

number

even:

1.35

=^>

1.4 1.25

=>

1.2

Express the following quantitiesin scientific notation

and

indicate

how many

significant figureseachhas.

(a) 12,200 (b) 12,200.0 (c) 0.003040

Expressthefollowing quantities instandarddecimal

form and

indicate

how many

signif- icant figureseachhas.

(a) 1.34

X

10⁵ (b) 1.340

x

^10~2 (c) 0.00420

X

10⁶

How many

significantfigures

would

thesolution ofeachof the following

problems have?

What

are the solutionsof(c)

and

(d)?

(a) (5.74)(38.27)/(0.001250) ^(c) 1.000

+

10.2 (b) (1.76

x

10⁴)(0.12

x

^10~6₎ (d) 18.76

-

7

Round

offeachofthefollowing

numbers

to threesignificantfigures, (a) 1465 (b) 13.35 (c) 1.765

x

^10~7

When

the value of a

number

isgiven, thesignificant figuresprovide an indication of the uncertaintyinthe value; forexample,avalueof2.7 indicates thatthe

number

lies

between

2.65

and

2.75.

Give

rangeswithin

which

eachof the following valueshe.

(a) 4.3 (d) 2500

(b) 4.30 (e) 2.500

x

10³ (c) 2.778

X

^10~3

2.5b Validating Results

Every problem you

willever

have

to solve

—

^inthis

^and

other courses

and

in

your

professional career

—

^willinvolve

^two

^criticalquestions:(1)

How do

Iget a solution?(2)

When

Igetone,

how do

I

know

^it'sright?

Most

ofthis

book

is

devoted

toQuestion¹

—

^thatis,to

^methods

^{of solving}

problems

that arise in the design

and

analysis ofchemicalprocesses.

However, Question

2 is

equallyimportant,

and

serious

problems

canarise

when

itisnotasked. All successfulengineers get into the habitofaskingit

whenever

they solve a

problem and

theydevelopa

wide

variety ofstrategiesforansweringit.

Among

approaches

you

can use to validate a quantitative

problem

solution are back- substitution, order-of-magnitudeestimation,

and

thetestofreasonableness.

• Back-substitutionisstraightforward:after

you

solveasetof equations, substitute

your

solu- tion

back

into theequations

and make

sureitworks.

• Order-of-magnitude estimation

means coming up

with acrude

and

easy-to-obtain approx- imation of the answer to a

problem and making

sure that the

more

exact solution

comes

reasonablyclosetoit.

•

Applying

the test ofreasonableness

means

verifying thatthe solution

makes

sense. It for example, a calculated velocity ofwaterflowing in apipe is fasterthan the

speed

oflightor the calculated temperaturein achemicalreactorishigherthanthe interiortemperature of the sun,

you

shouldsuspect that amistake has

been made somewhere.

The

procedureforchecking

an

arithmetic calculation

by

order-of-magnitudeestimation is

as follows:

1. Substitute simple integers forall numericalquantities,using

powers

of10 (scientificno- tation) forvery small

and

verylargenumbers.

27.36

—

* 20or30 (whichever

makes

thesubsequentarithmeticeasier) 63,472

—

6

X

10⁴

0.002887

->3X

^10~3

TEST

2.

Do

the resulting arithmetic calculations

by

hand, continuing to

round

off intermediate answers.

(36,720)(0.0624) ₍₄

X

10⁴)(5

X

1(T²₎

0.000478 5

X

^{10~4 4}

X

10(4_2+4

)

=

4

X

10⁶

The

correct solution (obtained using a calculator)is4.78

X

10⁶.If

you

obtainthissolution, sinceitisof the

same magnitude

asthe estimate,

you

can be reasonablyconfidentthat

you

haven't

made

a gross errorinthecalculation.

3. Ifa

number

is

added

to asecond,

much

smaller,

number, drop

thesecond

number

inthe

approximation. _^ ^.

— ₌ - =0

25 4.13

+JL04762 4

The

calculator solutionis0.239.

SOLUTION

Order-of-Magnitude Estimation

The

calculationofaprocess stream volumetric flowratehasled to the following formula:

V =

²⁵⁴

+

¹³

x

¹

.(0.879)(62.4) (0.866)(624)_J (31.3145)(60) Estimate

V

without usingacalculator.(Theexact solutionis0.00230.)

1 5

V »

^{250 Wf}

50

+

^

⁽⁴

^X

10^(6

X

10¹₎ 25

X

10²

*

0.2

X

lO"²

=

0.002

The

third

way

tocheckanumerical result—

and

perhapsthefirstthing

you

should

do when you

get

one—

^istoseeiftheanswerisreasonable.If,forexample,

you

calculate thata cylinder contains 4.23

x

10³²

kg

of

hydrogen when

the

mass

of the sunis only 2

x

₁₀³⁰ kg, it should motivate

you

to redo the calculation.

You

should similarly

be

concerned if

you

calculate a reactor

volume

largerthanthe earth (10²¹

m

³)ora

room

temperaturehot

enough

tomeltiron (1535°C). If

you

getinthe habit of askingyourself,

"Does

this

make

sense?" every time

you come up

with a solution to a

problem—

in engineering

and

in therest ofyour life

— ^you

^will

spare yourself considerablegrief

and

embarrassment.

2.5c Estimation of Measured Values: Sample Mean

Suppose we

carryoutachemicalreaction of the

form A

^-* Products, startingwithpure

A

ⁱⁿ

the reactor

and

keepingthe reactortemperatureconstantat45°C. After

two

minutes

we draw

a

sample from

the reactor

and

analyzeittodetermine

X,

thepercentageof the

A

^fedthathas reacted.

Coolant (fortemperature control)

In theory

X

should have a unique value; however, in a real reactor

X

^{is a}

^random

^variable,

changing in

an

unpredictable

manner from one

run toanotheratthe

same

experimentalcon-

2.5

Numerical

Calculation

and

Estimation 17 ditions.

The

values of

X

^{obtained after}10 successive runsmight beas follows:

Run

^{1 2 3 4 5 6 7 8 9 10}

X(%)

^{67.1 73.1 69.6 67.4 71.0 68.2 69.4 68.2 68.7 70.2}

Why

^don't

^we

^{get the}

^same

^{value of}

^{X'm each}

^run?

^There

^{areseveralreasons.}

• _Itisimpossibletoreplicateexperimentalconditions exactlyinsuccessiveexperiments.Ifthe temperature inthe reactorvariesby aslittle as 0.1 degree

from one

run toanother,itcould

be enough

tochange the

measured

^{value of}X.

•

Even

^ifconditions

^were

^identicalin

^two

^runs,

we

could not possibly

draw our sample

atex- actlyt

—

_2.000...minutes both^times,

and

a difference of asecond could

make

ameasurable difference inX.

• Variationsinsampling

and

chemical^analysisproceduresinvariablyintroducescatter in

mea-

sured^values.

We ^might

^ask

^two

^{questionsaboutthesystemat thispoint.}

1.

What

^isthetruevalue

of X?

In principle there

may

be such a thing as the "true

value"—

that is, the value

we would measure

^if

we

couldset thetemperature exactlyto45.0000^... degrees,startthe reaction,

keep

thetemperature

and

allotherexperimental^variablesthat affect

X

perfectly constant,

and

then

sample and

analyzewithcomplete^{accuracyatexactlyt}

=

2.0000...minutes. In practice there

is

no way

to

do any

ofthose things, however.

We

could alsodefine the true valueof

X

^as the value

we would

calculate

by

performing an infinite

number

of

measurements and

^averaging theresults,butthereis

no

^practical

way

to

do

thateither.

The

best

we

can ever

do

is toestimate the true value of

X from

afinite

number

of

measured

values.

2.

How

can

we

estimate

of

^{the truevalueof}

X?

The most common

^{estimate is thesample}

^mean

(or arithmetic

mean)

suredvalues of

X ^(X

x,

X

2,... ,

X _N

)

and

then^calculate

1

1^

Sample Mean: X ⁼ -(Xi + X

₂

+ + X N

)

= - _{2_}

^Xj

N

j=\

For

thegiven data,

we would

estimate

X ^{= JL(67.1%} + 73.1% +

^•••

+ ^{70.2%) = 69.3%}

Graphically, the data

and

sample

mean

might appear ^as

shown

below.

The measured

^values scatteraboutthe

sample mean,

^{asthey must.}

.

We

^collect

N mea-

(2.5-1)

X=69.3%

10 Run

The more measurements

of a

random

^variable, the better the estimated value based

on

the

sample mean. However,

^{evenwith a}

huge number

of

measurements

the

sample mean

isat best

an

approximationof thetruevalue

and

couldin factbe

way

off(e.g., ifthereissomething

wrong

withtheinstrumentsorprocedures used^to

measure

X).