SUMMARY

TEST

SOLUTION 1. Semilog plot

2.8 SUMMARY

This chapterintroduces

some

fundamental problem-solvingtoolsthat

you

will

need

intherest ofthis course,in subsequent engineering

and

science courses,

and

almost every time inyour career

when you perform

mathematical calculations.

The main

points of the chapter are as follows.

•

You

can convert a quantityexpressedin

one

set ofunitsinto itsequivalentin otherdimen- sionally consistentunitsusingconversion factors, likethose in the table

on

theinside front coverof thetext.

•

A

weightisthe force exerted

on

anobjectbygravitationalattraction.

The

weightofanobject of

mass m may

becalculatedas

W ⁼

mg,

where

_gisthe acceleration of gravityatthe location ofthe object.

At

sea level

on

theearth,g

=

9.8066Tin's2

=

32.174ft/s2

.

To

convertaweight (or

any

force)innaturalunitslikekg-m/s²orlb

m

^{-ft/s2toitsequivalent}ina derived forceunit like

N

orlb_{,usetheconversionfactortable.

•

The

significant figures(si.)withwhicha

number

isreportedspecify the precision withwhich the

number

^is

known.

Forinstance,x

=

_{3.0 (2} si.)states that

x

is

somewhere between

2.95

Problems

31

and

3.05,while

x =

3.000₍₄s.f.)statesthat itis

between

2.9995

and

3.0005.

When you

mul- tiply

and

divide numbers, the

number

of significant figures of the resultequals the lowest

number

ofsignificantfigures ofany ofthe factors. In

complex

calculations,

keep

the maxi-

mum number

^ofsignificant figures untilthefinalresultisobtained, then

round

off.

• If

X

^{is a}

^measured

^{process variable,}the

sample mean

ofa setof

measured

values,X, isthe averageoftheset(the

sum

of the valuesdivided

by

the

number

ofvalues).It is

an

estimate of the true

mean,

the valuethat

would

be obtained

by

averaging aninfinite

number

of

mea-

surements.

The sample

variance of the set, s

x

^{, is a}

^measure

^{of thespread of the}

^measured

valuesaboutthe sample

mean.

Itis calculated

from

Equation _(2.5-3).

The sample

standard deviation, s\,isthesquareroot of the

sample

variance.

• If

X and

s\ aredetermined

from

a setof

normal

process runs

and

asubsequently

measured

valueof

X

^falls

more

than 2s\

away from

X, the chancesarethat

something

has

changed

in theprocess

—

^thereislessthana

10%

chance that

normal

scattercanaccountforthe devia- tion.Ifthedeviationisgreaterthan3sx, there^islessthana 1

%

chancethat

normal

scatteris

the cause.

The

exactpercentages

depend on how

the

measured

values are distributedabout the

mean — ^whether

they followaGaussiandistribution,for

example — and how many

points are inthedatasetused tocalculate the

mean and

standarddeviation.

•

Suppose you

are givena set of values ofa

dependent

variable,y, corresponding to values of

an

independentvariable,

x

,

and you

wish toestimatey ^fora specified

x

.

You

caneither

assume

astraight-line

dependence

for the

two

datapoints thatbracket the specified

x and

use two-point linear interpolation (Equation 2.7-1) or fit afunction to the datapoints

and

use itfor thedesired estimation.

• If(x,y)dataappearto scatteraboutastraight line

on

aplotof

y

^{versus x,}

you may

fitaline usingEquations (2.7-3)

and

(2.7-4)or,forgreater precision

and

anestimate of thegoodness of thefit,use the

method

ofleastsquares

(Appendix

A.1).Ifaplotof

y

^versus

x

isnonlinear,

you may

try tofit various nonlinear functions

by

plottingfunctionsof

x and y

^{in a}

manner

thatshouldyieldastraightline.For example,tofita function y²

=

a/ x

+

bto(x_,y)data, plot y²versus l/x.Ifthefitisgood,the^plotshould

be

astraight linewith slopea

and

interceptb

.

• Plotting

y

(log scale) versus

x

(linear scale)

on

a semilogplotisequivalenttoplotting In

y

versus

x on

rectangular axes. Ifthe plot^is linear in either case,

x and y

are related

by an

exponentialfunction, y

=

ae^bx.

• Plottingy^versus

x on

logarithmicaxesisequivalenttoplotting In

y

^versusIn

x on

rectangular axes.Iftheplotislinear in either case,

x and y

are related

by

a

power

lawfunction,y

~

_ax^b.

PROBLEMS

2.1. Usingdimensionalequations,convert

(a) 3

wk

to milliseconds. (c) 554

m

⁴/(daykg) tocmV(inin-g).

(b) 38.1 ft/sto miles/h.

23.. Usingthe tableof conversionfactorsontheinside front cover,convert (a) 760miles/h to m/s. (c) 5.37

X

10³ kJ/minto hp.

(b) 921

kg/m

³tolbm^/ft3.

23. Using a single dimensional equation, estimate the

number

of golf ballsit

would

take to fillyour classroom.

Usinga singledimensionalequation,estimatethe

number

of stepsitwouldtakeyou, walkingatyour normalstride,towalkfromtheEarthto

Alpha

Centauri,a distance of4.3 light-years.

The

speedof lightis 1.86

X

10⁵ miles/s.

A

^{frustratedprofessoronceclaimedthatifallthereports}shehad gradedinher careerwerestacked

on

top ofoneanother, theywouldreachfromtheEarthto the

moon. Assume

thatanaveragereport

isthethicknessof about 10sheetsof printerpaperandusea singledimensional equationtoestimate the

number

of reports the professorwould have had togradeforher claimtobevalid.

You

aretrying todecidewhichoftwoautomobilesto buy.

The

firstisAmerican-made,costs $14,500,

and

hasa rated gasolinemileageof28miles/gal.

The

secondcarisof

European

manufacture,costs

$21,700,andhasaratedmileage of 19km/L.Ifthe cost of gasolineis$1.25/galandifthe cars actually deliver their rated mileage, estimate

how many

milesyou

would

have todrive for thelower fuel consumptionofthesecondcar tocompensateforthehighercostofthiscar?

2.7.

A

supersonic aircraft consumes 5320 imperial gallons ofkerosene per hour offlightand flies an average of 14 hours perday.Ittakesroughlyseventonsofcrudeoiltoproduceoneton of kerosene.

The

density ofkeroseneis0.965g/cm³.

How many

planes

would

ittaketo

consume

the entireannual worldproduction of4.02

X

10⁹metrictonsofcrude oil?

2.8. Calculate

(a) theweightinlbfof a 25.0-lb

m

^object.

(b) themassinkgofanobject thatweighs25 newtons.

(c) theweightindynes of a10-ton object(notmetrictons).

A

waste treatment

pond

^is50

m

longand15

m

wide,andhasanaveragedepthof2m.

The

densityof the waste^is85.3lbm^/ft3.Calculatetheweightof the

pond

contentsin lbf,usinga singledimensional Student equationforyourcalculation.

2.10. Fivehundredlbm^{ofnitrogenistobe}chargedintoasmallmetalcylinderat25°C,atapressure such thatthegasdensityis11.5

kg/m

³.Withoutusinga calculator,estimate the requiredcylindervolume.

Show

yourwork.

2.11. According to Archimedes'principle, themassofa floating objectequals the massofthe fluid dis- placedbytheobject.

Use

^thisprinciple to solve thefollowingproblems.

(a)

A ^wooden

cylinder 30.0

cm

highfloatsvertically ina tub ofwater(density

=

1.00g/cm³).

The

topofthecylinderis14.1

cm

abovethesurfaceof theliquid.

What

isthe densityofthe

wood

⁹

(b)

The same

cylinder floats vertically in a liquid of

unknown

density.

The

top of the cylinder is

20.7

cm

abovethesurfaceof theliquid.

What

isthe liquiddensity?

2.12.

A

right circularconeofbaseradius R,heightH, and

known

densityp^{s floatsbase}

down

ina liquid of

unknown

densitypf^.

A

height h of the cone is above theliquid surface. Derive a formulafor pf interms of p^{s, R,} and h/H, simplifying it algebraically to the greatest possible extent. [Recall Archimedes'principle,statedinthepreceding problem,andnotethatthevolumeof aconeequals (base area)(height)/3.]

2.13.

A

horizontalcylindrical

drum

is2.00

m

^indiameterand4.00

m

^long.

The drum

isslowlyfilledwith benzene(density

=

0.879g/cm³). DeriveaformulaforW, theweightinnewtonsofthebenzenein the tank,asafunctionofh,thedepthofthe liquidincentimeters.

2.14.

A

^poundalisthe forcerequiredtoaccelerate amassof1lb

m

ata rateof1 ft/s2

,anda slugisthemass ofanobject thatwillaccelerateat a rateof1ft/s2

when

subjectedtoa force of1 lbf^.

(a) Calculatethemassinslugsandtheweightinpoundals ofa175lbm

man

(i)onearthand(ii)on the

moon,

wheretheaccelerationofgravityisone-sixthofitsvalue

on

earth.

(b)

A

force of355 poundals isexerted

on

a 25.0-slug object.

At

whatrate (m/s²) doesthe object accelerate?

2.15.

The

fernisdefinedasthe unitofforcerequiredto accelerate a unitof mass,called thebung, withthe gravitational accelerationonthesurfaceof the

moon,

whichisone-sixth of thenormalgravitational acceleration

on

earth.

(a)

What

istheconversionfactor that

would

be usedtoconvert aforcefrom the natural unittothe derivedunitin thissystem? (Givebothitsnumerical valueanditsunits.)

(b)

What

istheweightinferns ofa3-bungobjectonthe

moon? What

does^the

same

objectweigh

inLizardLick,NorthCarolina?

2.16. Performthefollowing calculations.Ineachcase,firstestimatethe solutionwithout usingacalculator, following theprocedure outlined in Section2.5b,and thendo thecalculation, paying attention to significant figures.

(a) (2.7)(8.632) (c) 2.365+125.2

(b) (3.600

x

^10"4_)/45 (d) (4.753

x

10⁴)

-

₍₉

_x

₁₀²

)

Problems

₃₃

2.17.

The

following expression has occurredinaproblemsolution:

(0.6700)(264,980)(6)(5.386

X

10⁴₎

R =

(3.14159)(0.479

X

10⁷₎

Equipment

•

Encyclopedia thermocouple

The

factor 6 is a pure integer. Estimate the value of

R

without using a calculator, following the procedureoutlinedinSection2.5b.

Then

calculate R,expressingyour answer inbothscientificand decimal notationand

making

sureithasthe correct

number

ofsignificant figures.

2.18.

Two

thermocouples(temperature

measurement

devices)aretestedbyinsertingtheirprobesin boil- ing water, recording the readings, removing and drying the probes, and then doing it again.

The

resultsoffivemeasurementsareasfollows:

r(°C)—

Thermocouple

A

72.4 73.1 72.6 72.8 73.0 r(°C)

—

Thermocouple

B

97.3 101.4 98.7 103.1

I

100.4

(a) For each setof temperature readings, calculate the sample mean, the range, and the sample standarddeviation.

(b)

Which

thermocouplereadingsexhibitthehigherdegree ofscatter?

Which

thermocoupleis

more

accurate?

2.19. Productqualityassurance

(QA)

isa particularly tricky businessinthedye manufacturingindustry.

A

^{slightvariationinreaction}conditions can lead toa measurable changeinthe color of theprod- uct,andsincecustomersusuallyrequireextremely highcolor reproducibilityfrom one shipmentto another,evena smallcolorchangecanlead to rejectionof a productbatch.

Supposethevariouscolorfrequencyandintensityvaluesthatcomprisea color analysis arecom- binedintoa singlenumericalvalue,C,foraparticularyellowdye.Duringatestperiodinwhichthe reactor conditions arecarefullycontrolledandthereactoristhoroughly cleanedbetweensuccessive batches(nottheusualprocedure),product analysesof 12 batchesrun on successive days yield the followingcolor readings:

Batch 1 j 2 3 4 5 6 7 8 9 10 11 12

C

^{74.3 71.8 72.0} 73.1 75.1 72.6 75.3 73.4 74.8 72.6 73.0 73.7

(a)

The QA

specification forroutineproductionis that a batchthat falls

more

than two standard deviations

away

fromthetestperiod

mean must

berejectedandsent forreworking.Determine the

minimum

and

maximum

acceptable values ofC.

(b)

A

statistician workinginqualityassurance

and

aproduction engineer are havinganargument.

One

^ofthem, Frank,wants to raisethe

QA

specificationto three standard deviationsand the other,Joanne,wants toloweritto one.

Reworking

istime-consuming, expensive,andvery un- popular withtheengineers

who

havetodoit.

Who

is

more

likelytobethestatisticianand

who

theengineer?Explain.

(c) Suppose thatinthefirst fewweeksofoperationrelatively fewunacceptable batches are pro- duced, but thenthe

number

beginsto climb steadily. Thinkof

up

to five possible causes, and

state

how you

might go aboutdeterminingwhetherornoteachof

them

mightinfactbe respon- sibleforthedropin quality.

*2.20.

Your company

manufacturesplasticwrapforfoodstorage.

The

tear resistanceof the wrap,denoted byX,mustbecontrolledsothatthewrapcan be tornofftherollwithout too

much

effortbutitdoes notteartooeasily

when

inuse.

In a series of test runs, 15 rolls of

wrap

are

made

undercarefully controlled conditions and thetear resistanceof eachrollismeasured.

The

resultsare usedas the basis of a quality assurance specification (see Problem 2.19). If

X

for asubsequently produced roil fails

more

than two stan- darddeviations

away

from the testperiod average, the processisdeclared outof specification and productionissuspendedfor routinemaintenance.

*Computerproblem.

The

test seriesdataare asfollows:

Roll 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

X

134 131 129 133 135 131 134 130 131 136 129 130 133 130 133

2.21.

(a) Writea spreadsheettotakeasinputthetest seriesdataandcalculatethesample

mean

(X) and sample standard_deviation(jx),preferably usingbuilt-infunctions forthecalculations.

(b)

The

followingtear resistancevalues are obtainedforrollsproducedin14 consecutive production runssubsequentto thetest series: 128, 131, 133, 130, 133, 129, 133, 135, 137, 133, 137, 136, 137, 139.

On

^the spreadsheet (preferably usingthe spreadsheet plotting capability), plot a control chartof

X

versusrunnumber, showinghorizontallinesforthevaluescorrespondingtoX,

X ^-

2sx,and

X +

2sxfromthetestperiod,and

show

the pointscorrespondingtothe 14 production runs.(See Figure2.5-2.)

Which

measurementsled tosuspension of production?

(c) Followingthelastof theproductionruns,thechief plantengineerreturns

from

vacation,exam- inestheplantlogs, andsays that routine maintenance wasclearlynotsufficient and a process

shutdown

andfullsystem overhaul shouldhave beenorderedatonepointduringthetwo weeks hewasaway.

When

wouldithave beenreasonable totakethis step,and

why?

A

^variable,_Q,isreportedtohavea value of 2.360

X

^10~4

kg-m

²/h.

(a) Writea dimensional equation forQ',the equivalentvariable value expressedin

American

en- gineeringunits,usingsecondsastheunit for time.

(b) EstimateQ' without usinga calculator,followingtheprocedure outlinedinSection2.5b.

(Show

yourcalculations.)

Then

determineQ'with acalculator,expressingyouranswerinbothscientific

anddecimal_notationand

making

sureithasthe correct

number

ofsignificant figures.

2.22.

The

Prandtl number,

N

Pr, is a dimensionless group important _in heat transfer calculations. It is

defined as

C>/

^k, where

C

_p is the heat _capacity ofa fluid, p. is the fluid viscosity, and

k

is the thermal conductivity. For a particularfluid,

C

p

=

0.583 J/(g-°C), k

=

_0.286 W/(m-°C), and p.

=

1936 ^lb

m

^/(ft-h). Estimate the value ofjV_Pr withoutusing a calculator (remember, it is dimension- less),showing yourcalculations;then determineitwithacalculator.

2.23.

The

Reynolds

number

isadimensionlessgroupdefinedfor afluidflowingina pipeas

Re = Dup/

p,

where D

ispipe diameter, uisfluid velocity,pisfluid density,and

p

^isfluid viscosity.

When

the value of theReynolds

number

islessthanabout2100,theflowis

laminar—

thatis,thefluidflowsin

smooth

streamlines. For Reynolds

numbers

above2100, theflowisturbulent,characterizedbyagreat deal ofagitation.

Liquidmethylethylketone

(MEK)

flowsthrough a pipe with an innerdiameterof 2.067 inches at an average velocity of 0.48ft/s.

At

the fluid temperature of

20°C

the density ofliquid

MEK

^is

0.805g/cm³ and theviscosityis0.43 centipoise _[1cP

=

1.00

X

^10"3 _{kg/(m-s)]. Withoutusinga cal-}

culator,determine whetherthe flowislaminar orturbulent.

Show

yourcalculations.

2.24.

The

followingempiricalequationcorrelatesthe values ofvariables_inasysteminwhichsolidparti- cles aresuspendedinaflowinggas:

k9

d

_py

D ⁼

^2.00

⁺

^0.600

^{f-^} \pD

1/3 d„up ^1/2

Both

(fx/

pD)

and(d_pup/_p.) aredimensionless groups; k

g isa coefficient thatexpresses the rateat

whichaparticular species transfersfromthegasto thesolid particles;and thecoefficients 2.00and 0.600 are dimensionless_constants obtainedbyfitting experimental data covering a widerange of values of the equationvariables.

The

value of k_gisneededtodesignacatalytic reactor.Sincethiscoefficientisdifficultto deter-

mine

^directly,values of theothervariablesaremeasuredorestimatedand k_giscalculatedfromthe givencorrelation.

The

variablevaluesare as follows:

d_p

=

5.00

mm

y

=

0.100 (dimensionless)

Problems

35

D ⁼

0.100

cm

²/s fi

=

1.00

X

^10"5 N-s/m²

p

=

1.00

X

^10~3 g/cm³

«

=

10.0m/s (a)

What

istheestimated value ofk

gl (Giveitsvalueandunits.)

(b)

Why

mightthe truevalue of k_ginthe reactorbesignificantlydifferentfromthevalue estimated inpart (a)? (Giveseveral possible reasons.)

*(c) Create a spreadsheetinwhich uptofivesetsof values ofthegivenvariables (d

p throughu)are enteredincolumns andthe corresponding values ofk_gare calculated.Testyour

program

using the following variable sets: (i) the values given above; (ii) as above, only double the particle diameter d_p (making it10.00

mm);

(iii) asabove, only doublethe diffusivityD; (iv) as above, only doubletheviscosity (v) asabove, onlydoublethe velocityu.Reportallfivecalculated valuesof k

g^.

A

^seed crystalof diameter

D (mm)

is placedin a solution of dissolved salt, and

new

crystalsare observedto nucleate (form) at aconstant rate r (crystals/min). Experiments withseed crystalsof differentsizes

show

thattherate of nucleation varieswith theseedcrystaldiameteras

r(crystals/min)

= _200D - _10D

² _(£>in

mm)

(a)

What

are theunitsofthe constants200and 10?

(Assume

thegiven equationisvalidand there- foredimensionallyhomogeneous.)

(b) Calculate the crystal nucleationrate in crystals/s correspondingto a crystal diameter of0.050 inch.

(c) Deriveaformulaforr(crystals/s) interms of D(inches). (See

Example

2.6-1.)

Check

theformula using the resultofpart(b).

The

densityof afluidisgivenbythe empiricalequation p

=

70.5exp(8.27

X

^10"7P)

where

_pisdensity(lb

m

^/ft3) and

P

ispressure(lb_f/in.²_).

(a)

What

arethe unitsof70.5 and8.27

X

^10"7?

(b) Calculate thedensitying/cm³ forapressure of9.00

x

10⁶

N/m

^2.

(c) Deriveaformulaforp(g/cm³)asa functionof

P(N/m

²).(See

Example

2.6-1.)

Check

yourresult using the solutionofpart(b).

2.27.

The volume

of amicrobialcultureisobservedtoincrease accordingtotheformula

V(cm

³)

=

_e^!

where

f istimeisseconds.

(a) Calculate theexpressionfor V(in.³₎interms off(h).

(b)

Both

the exponential function and its argument must be dimensionless.

The

given equation seems to violatebothoftheserules,andyet theequationisvalid. Explain thisparadox. [Hint:

Observethe resultofpart(a).]

2.28.

A

concentration

C

(mol/L)varieswith time (min) accordingtotheequation

C =

3.00 exp(-2.00?) (a)

What

arethe units of 3.00and2.00?

(b) Supposethe concentration ismeasuredatt

=

0and t

=

1 min.

Use

two-pointlinearinterpo- lationor extrapolationto estimate C(f

=

0.6min) andt(C

=

0.10mol/L)

from

the measured values,and

compare

theseresultswith the truevalues ofthese quantities.

(c) Sketchacurve of

C

versust,and

show

graphically the pointsyou determinedinpart(b).

*2.29.

The

vaporpressures of 1-chlorotetradecaneatseveraltemperaturesaretabulatedhere.

T(°C)

98.5 131.8 148.2 166.2 199.8 215.5

p

^*

(mm Hg)

1 5 10 20 60 100

*Coraputer problem.

(a)

Use

two-pointlinear interpolation toestimate the valueofp*^at

T =

_185°C.

(b) Write a computer subroutine to estimate the vapor pressure of 1-chlorotetradecane for any temperature between98.5°Cand 215.5°Cusing two-pointlinearinterpolation.

The

subroutine must determine which two tabulated temperatures bracket the given temperature, andapply the interpolation to estimate

p*

^(T).

Then

write a

main

program to read and store the val- ues of p* and

T

givenin the tableand to generate a table ofvapor pressuresat temperatures

T =

100°C, 105°C,110°C, ...,215°C, callingyoursubroutine toestimate p* at each tempera- ture.

Check

yourprogramusing theresultofpart(a).

2.30. Sketchtheplotsdescribedbelow andcalculate the equationsfory(x)fromthe given information.

The

plotsareallstraightlines.

Note

thatthe givencoordinatesrefertoabscissaandordinatevalues, not

x

and_y values. [Thesolutionofpart(a)isgivenasanexample.]

(a)

A

^{plot oflny versus}x on rectangular coordinates passes through (1.0, 0.693) and _{(2.0, 0.0)}

(i.e.,atthefirstpointx

=

1.0andIny

=

_0.693).

Solution: Iny

=

bx

+

Ina

=>

y

=

ae^bx

b

=

(lny₂

-

\nyl)/{x₁

-

x{)

=

₍₀

-

_0.693)/(2.0

-

_1.0)

=

-0.693

Ina

=

_lnvi

-

bxx

=

0.693

+

0.693* _1.0

=

1.386

=>

a

=

e

im ₌

4.00

I

y

=

_4.00e'°-^693x

(b)

A

semilogplotof y (logarithmicaxis)versusxpassesthrough_(1,2)and(2, 1).

(c)

A

^{log plot}of y ^versusx passesthrough _(1,2)and(2, 1).

(d)

A

semilogplot ofxy(logarithmicaxis)versusyj x passes through(1.0,40.2)and(2.0,807.0).

(e)

A

^{log plot}_{of y}^2/xversus (x

-

_{2) passesthrough(1.0,40.2)and(2.0,807.0).}

231. Statewhat you wouldplottoget a straightlineifexperimental_{(x, y)}dataare tobecorrelatedbythe followingrelations,and whatthe slopesandintercepts

would

beinterms ofthe relationparameters.

If

you

could equally welluse twodifferentkinds ofplots (e.g., rectangular or semilog),statewhat you

would

plotineachcase. [Thesolution to part(a) isgivenasan example.]

(a) y²

=

ae~^b-/x.

Solution:Constructasemilogplotof y²versus \j x ora plotofIn(y²)versus llx

on

rectangular coordinates. Slope

=

-b,intercept

=

Ina.

(b) y

2

=

rax³

—

n

(c) 1/ln(y

-

₃₎

=

₍₁

+

a

fx)/b

(d) (y

+

l)²

=

[a(x

-

3)³]"¹ (e) y

=

exp(a

Jx ⁺

^b)

(f)

xy = WW*

²

(g) y

=

[ax

+

b/x]-¹

2.32.

A

hygrometer,which measuresthe

amount

ofmoistureinagas stream,istobecalibratedusing the apparatus

shown

here:

Student Workbook

Air

Water