SUMMARY

CREATIVITY EXERCISES

3.6 SUMMARY

Temperature Conversion and Dimensional Homogeneity

The

heat capacityof

ammonia,

defined asthe

amount

ofheat required to raisethe temperatureof aunitmassof

ammonia

byprecisely1° ataconstantpressure,is,overa limitedtemperaturerange, givenbythe expression

Rtn \

0.487

+

2.29

X

10^_4r(°F)

^Ub

m-°F

Determine

theexpressionfor

C

_pin J/(g^•°C)intermsof T(°C).

SOLUTION The

°Finthe unitsof

C

_p refers to atemperatureinterval,whilethe unit of

T

isatemperature.

The

calculationisbestdoneintwosteps.

1. Substitute for7(°F)andsimplifythe resultingequation:

Btu

p! iK or I

-1 °-487

+ 229 X

10

[1.87TO +

32]

=

0.494

+

4.12

X

10^_47(°C)

2. Convertto the desiredtemperatureinterval unitusingEquation^3.5-5:

J g-°c

=

[0.494

+

4.12

X

^10"4r(°C)]-

I

(Btu) 1.8°F

1J

_{llb m}

(lb

m

^{-°F) 1.0°}

C

9.486

X

^10"4 Btu 454_g

g-°c

=

2.06

+

1.72

X

10^_jr(°C)

CREATIVITY EXERCISES

1. Inventseveral temperature-measuringdevices. For each, describe the device, and statewhat

you would

measure. {Example:Putaguineapigonatreadmillina

room

and measurethe rate atwhich herunstokeep warm.) (Well,itcouldwork.)

2. Think of as

many

ways as you can to use a solid block as a temperature-measuring device.

(Example: Put itin a furnace equipped with a

window

and observe the color with which it

glows.)

3.6

SUMMARY

Inthischapter,

we

havedescribed

how amounts

of material, flowrates,compositions, pressures,

and

temperaturesaredetermined

from

direct

measurements

or calculated

from measurements and

physicalproperties.

We

^alsohave described

how

to convert

between

different

methods

of expressingthesevariables.

Here

are

some

highlights.

•

The

densityof asubstanceis theratio ofits

mass

toitsvolume. For example,the density of liquidacetoneat20°

C

^is0.791

g/cm

³,so that

one

cubic centimeter ofliquidacetoneat20°

C

hasa

mass

of 0.791gram. Density can be thoughtofas aconversionfactor

between mass and volume

^or

between mass

flowrate

and

volumetricflowrate.

•

The

specific gravityof asubstanceistheratioof the density of thesubstanceto the density of a reference material(typicallywaterat4°C).Specific gravitiesof

many

liquids

and

^solidsare givenin TableB.l, with the reference densitybeing thatofliquidwater at

4°C

(1.00

g/cm

³,

1.00 kg/L, 62.43 ^lb

m

^/ft3).

^The

^{densityofa substance}istheproductofits specificgravity

and

the reference densityin the desiredunits.

•

The

atomicweight of

an

element is the

mass

ofan

atom

ofthat

element on

ascale that as- signs ¹²

C

^a

^mass

^{of exactly12.}

Atomic

weightsof the elementsin theirnaturally occurring

isotopicproportionsarelisted inthetable atthe

back

of thebook.

The

molecularweight of a

compound

isthe

sum

of theatomicweights of the

atoms

thatconstitute amolecule ofthat

compound.

gram-mole

or

mol

of a

compound

isthemolecular weightof the

compound

ingrams;for example, 1

mol H

2

0

^{has a}

mass

of18.01 grams.

A pound-mole

or lb-mole isthe molecular weightin

pounds

mass;forexample, 1lb-mole

H

2

0

has a

mass

of 18.01 lb

m

^.

^The

^molecular

weightofwater

may

thereforebe expressedas 18.01 g/mol, 18.01lb

m

^/lb-mole,

^and

^{so on,}

and

it

may

be used to convert masses tomoles or

mass

flow rates to

molar

flow rates

and

vice versa.

•

The mass

fraction of a

component

inamixture isthe ratioof the

mass

of the

component

to the total

mass

ofthe mixture. If100

grams

of amixture contains30

grams

of nitrogen, the

mass

fraction ofnitrogen is0.30 g

N

₂/gmixture. (The

word

"mixture" usually is omitted.)

The mass

fraction is also 0.30 kg

N

₂/kg

and

0.30 lb

m N

2/lb

m

^,

^and

the percent by

mass

or weight percent of nitrogenis

30%. The mole

fraction ofa

component

is

denned

similarly. If 10.0

kmol

of a mixture contains 6.0

kmol

of methanol, the

mole

fraction of

methanol

is

0.60

kmol CH

3

OH/kmol (=

0.60 lb-mole

CH

₃OH/lb-mole),

and

the

mole

percent of

methanol

is

60%.

•

The

average molecular weightofamixtureistheratio of thetotal

mass

to thetotal

number

of

moles

ofallspecies.

•

The

concentration of a

component

inamixtureistheratioof the

mass

or

moles

of the

com- ponent

^to the totalmixture volume.

The

molarity of a

component

of a solutionis the con- centration of the

component

expressedinmol/L.

•

The

pressureatapointinafluid(gasorliquid)isthe forceperunitareathatthefluid

would

exert

on

aplanesurface passingthroughthepoint.Standardunitsoffluidpressure are

N/m

²,

(pascal,orPa) inthe SI system,

dyne/cm

² inthe

CGS

system,

and

lb_{f /ft}²inthe

American

en- gineering system.

The

unit lb_f/in.2

(psi)isalso

common

inthe

American

engineeringsystem.

•

The

pressureatthebaseof avertical

column

offluidof density

p and

height

h

isgivenbythe expression

P = P

_Q

+ pgh

_(3.4-1)

where P

₀isthepressureexerted

on

thetop of the

column and

_g isthe acceleration ofgravity.

This result gives rise to

two ways

of expressingfluid pressure: as force per unit area (e.g.,

P =

14.7 lbf/in.

2

) oras

an

equivalent pressure head,

P

_h

= Pj _pg

(e.g.,

P

_h

=

₇₆₀

mm

Hg), the height ofa

column

of thespecifiedfluidwith zero pressure atthe top that

would

exert thespecifiedpressureatthebottom.

•

The

earth's

atmosphere

can beconsidered a

column

offluidwith zeropressureatthetop.

The

fluidpressureatthebase ofthis

column

isatmosphericpressureor barometricpressure,

PMm

^.

Although

atmosphericpressurevarieswithaltitude

and weather

conditions,itsvalue atsea levelisalwaysclose to1.01325

x

10^s

N/m

²

(=

^14.696lb_f/in.2

=

760

mm

Hg).This pressure value has

been

designated 1 atmosphere.

Other

equivalents to 1

atm

in different units are given

on

theinsidefrontcoverofthis text.

•

The

absolute pressure of a fluidis the pressurerelative to a perfect

vacuum (P =

_0).

_The gauge

pressureisthepressurerelative toatmosphericpressure:Pgauge

= P

_{a bs}

^~ Com- mon

pressure gauges like a

Bourdon

gauge

and an open-end manometer

provide a direct readingof

gauge

pressure.Ifatmosphericpressureisnot

known from

a

weather

report or a

barometer

reading, avalue of

P

_atm

=

1

atm

isusuallyreasonableto

assume when

converting

between

absolute

and gauge

pressure.

• Temperaturescalesareobtained byassigningnumericalvalues to

two

experimentallyrepro- ducibletemperatures.For example,the Celsiusscale isobtained

by

assigninga value of

0°C

tothe freezing point ofpure waterat1

atm and

a value of 100°

C

tothe boiling point ofpure waterat1atm.

A

temperatureof

40°C

isconsequently shorthandfor "thetemperature

40%

of the

way from

the freezing point ofwaterat 1

atm

tothe boiling point ofwaterat 1 atm."

•

The

four

most common

temperature scales are Celsius (°C), Fahrenheit (°F),

and

the ab- solutetemperature scalesKelvin (K)

and Rankine

(°R).

Temperatures

expressedin

one

of

Problems

65

Interactive Tutorial #1

Questions with Immediate Feedback

thesescales

may

be converted toequivalenttemperaturesinanotherscale usingEquatiions 3.5-1 through3.5-4.

Temperatures

should not be confused with temperatureintervals. For example,_a tempera- tureof

10°C

^isequivalenttoatemperatureof

50°F

(from

Equation

3.5-4),butatemperature intervalof

10°C

(e.g.,theinterval

between T = 10°C and T =

_20°C)isequivalenttoatem- perature interval of 18°F (the interval

between 50°F and

68°F).

An

interval of 1 Celsius degree or 1 Kelvinisequivalent to1.8Fahrenheitor

Rankine

degrees.

Note: This

would

bea

good

time to

work

through thefirstinteractive_tutorial.

PROBLEMS

3.1. Performthefollowing estimations withoutusingacalculator.

(a) Estimatethemassof water(kg)inan Olympic-size

swimming

pool.

(b)

A

^{drinkingglassisbeingfilled}froma pitcher.Estimatethemassflowrate of thewater(g/s).

(c) Twelve heavyweight boxerscoincidentallyget

on

the

same

elevatorinGreatBritain.Postedon the elevatorwallisa sign that gives the

maximum

safe

combined

weight of the passengers,

W

max,

instones₍₁stone

=

14lbm

«

6kg). Ifyou were oneofthe boxers,estimate the lowest value of

W

m3X forwhich you wouldfeelcomfortableremaining

on

the elevator.

(d)

An

oilpipeline across Alaskais4.5ftindiameterand 800miles long.

How many

barrelsofoil

are requiredto fillthe pipeline?

(e) Estimatethe

volume

ofyourbody (cm³)intwodifferent ways.

(Show

yourwork.)

(f)

A

solid block is droppedinto waterandvery slowly _{sinks to}the bottom. Estimate itsspecific gravity.

3.2. Calculatedensitiesin lbm/ft³ofthe following substances:

(a) a liquidwithdensityof995kg/m³.

Use

(i)conversionfactors fromthe table ontheinside front coverand(ii) Equation3.1-2.

(b) asolidwithaspecificgravity of5.7.

3.3.

The

specificgravity of gasolineisapproximately0.70.

(a) Determinethemass(kg) of 50.0litersofgasoline.

(b)

The

massflowrate ofgasoline exiting a refinery tank is 1150 kg/min. Estimate the volumetric flowrateinliters/s.

(c) Estimate the average massflowrate (lbm/min) deliveredbya gasoline

pump.

(d) Gasolineand kerosene(specificgravity

=

0.82) areblended toobtainamixture with aspecific gravityof0.78.Calculate thevolumetricratio (volumeof gasoline/volume of kerosene) ofthe two

compounds

inthe mixture,assuming

V

b|end

= V

gasoline

+ V

kerosene^.

3.4.

Assume

thepriceofgasolineinFrance isapproximately5Frenchfrancsperliter

and

theexchange rateis 5.22 francs per U.S. dollar.

How much would you

pay, in dollars,for 50.0 kgof gasoline in France,assuminggasoline hasaspecificgravityof0.70.

What would

the

same

quantity ofgasoline costin theUnitedStatesata rateof$1.20per gallon?

3.5. Liquid benzene and liquid n-hexane are blended to form a stream flowing ata rate of700 lbm^/h.

An

on-line densitometer (an instrument usedtodeterminedensity) indicates thatthestreamhasa density of 0.850g/mL. Usingspecific gravitiesfromTableB.l,estimatethemass andvolumetric feed rates of the two hydrocarbonsto themixing vessel(in

American

engineering units). State at least

twoassumptions requiredtoobtaintheestimate fromthe

recommended

data.

3.6.

At

25°C,anaqueoussolutioncontaining35.0

wt% H

2

S0

4 has aspecificgravityof1.2563.

A

quantity of the

35%

^solutionisneededthatcontains 195.5kgof

H

2

S0

4^.

(a) Calculate the required volume(L) of the solutionusing thegivenspecific gravity.

(b) Estimate the percentageerrorthatwould haveresultedifpure-componentspecificgravities of

H2SO4 (SG =

1.8255)andwaterhad beenusedforthe calculation instead ofthe givenspecific gravityofthemixture.

3.7.

A

rectangular block of solid carbon (graphite) floats at the interface of two immiscible liquids.

The

bottom liquid is a relatively heavy lubricating oil, and the top liquid is water.

Of

the ^total block volume,

54.2%

is immersed in the oil and the balance is in the water. In a separate ex- periment, an

empty

flask is weighed, 35.3

cm

³ of the lubricating oil is poured into the flask, and the flask is reweighed. If the scale readingwas 124.8 g ^{in the first} weighing,

what would

it be in the second weighing? (Suggestion: Recall Archimedes' ^principle, and do a force balance

on

the block.)

Student Workbook

Student Workbook

Equipment Encyclopedia

ball mill, aka tumblingmill

3.8.

A

rectangular block floats in pure waterwith0.5 in. above the surface and 1.5 in. below the sur- face.

When

placed inan aqueoussolution,the block of material floatswith 1in. belowthesurface.

Estimatethe specificgravitiesofthe blockand the solution. (Suggestion: Callthehorizontal cross- sectionalareaoftheblockA.

A

should cancelinyourcalculations.)

3.9.

An

object ofdensitypa,volume

V

_a,andweight

W

t isthrown fromarowboatfloating

on

the surface ofasmall

pond

andsinks to thebottom.

The

weight of therowboatwithout thejettisoned objectis

W

b^. Before theobjectwas thrownout, the depthofthe

pond was

h_pl, andthe

bottom

of the boat wasa distance h

M

^abovethe

^pond

bottom. Afterthe objectsinks,the values of thesequantities are h_p2 andh_b2.

The

area ofthe

pond

is

A

p^;that oftheboatis

A

b^.

A

_b

may

be

assumed

constant, so that the

volume

ofwaterdisplacedbythe boatis

A

_b(h

p

-

_hb).

(a) Derive anexpressionforthe changeinthe

pond

depth(h_p2

-

_h

pi).

Does

theliquidlevelof the

pond

riseorfall,orisitindeterminate?

(b) Derivean expressionforthechangeintheheight of thebottomof theboatabovethebottomof the

pond

(h_b2

-h

bl).

Does

theboatriseorfallrelativetothe

pond

bottom, oris itindeterminate?

Limestone(calcium carbonate)particlesare storedin50-Lbags.

The

voidfractionof theparticulate matteris0.30 (literofvoid space perliteroftotalvolume) andthespecificgravityofsolidcalcium carbonateis2.93.

(a) Estimatethebulkdensityofthebagcontents (kg

CaC0

3/literoftotalvolume).

(b) Estimatetheweight(IV) of thefilledbags.Statewhat

you

are neglectinginyourestimate.

(c)

The

contents of three bags are fed to a ball mill, a device something like a rotating clothes dryer containingsteelballs.

The

tumblingactionoftheballscrushes the limestoneparticlesand turns

them

_{into a powder. (See}

pp. 20-31 ofPerry's ChemicalEngineers'

Handbook,

7th ed.)

The

limestonecomingout ofthe millisputbackinto50-Lbags.

Would

thelimestone(i)justfill

three bags, (ii)fallshort offillingthreebags,or (iii)fill

more

than threebags? Briefly explain youranswer.

A

^useful measureofanindividual's physical conditionis thefraction ofhis or her

body

thatcon-

sistsoffat.This problemdescribesa simple techniqueforestimatingthis fraction

by

weighingthe individual twice,oncein airand once

submerged

inwater.

(a)

A man

has

body

mass

m

b

=

122.5 kg. If he stands

on

ascale calibrated to read in newtons, what

would

the reading be? Ifhethen standsonascalewhileheistotally

submerged

inwater at 30°

C

(specificgravity

=

0.996) and the scale reads44.0 N, whatis the

volume

ofhisbody

(liters)?(Hint:Recallfrom Archimedes'principle that theweight of a

submerged

objectequals theweightin airminusthebuoyantforceonthe object,whichinturnequals the weightofwater displacedbythe object.Neglect thebuoyantforce ofair.)

What

ishis

body

density, pb (kg/L)?

(b) Suppose the

body

is divided into fat and nonfat components, and that

X

f (kilograms of fat/kilogramsoftotal

body

mass)isthefractionof thetotal

body mass

thatisfat:

3.11

x_f

= —

m

_b

Provethat

J_

_ J_

r

=

^{Pb Pnf}

I- J_

Pf Pnf

where

_pb,pf,and_pnfarethe averagedensitiesofthewholebody,thefatcomponent, andthenon- fatcomponent,respectively.[Suggestion: Startbylabeling themasses

(m

fand m„) and volumes

(V

₍and

V

_b)ofthefat

component

of the

body

andthe wholebody, andthen writeexpressions forthe three densities in terms ofthese quantities.

Then

eliminate volumes algebraicallyand obtainanexpressionfor

m

{/m_binterms of thedensities.⁵ ]

(c) Iftheaverage_{specificgravity}of

body

fatis0.9andthatofnonfattissueis1.1,

what

fraction of theman's

body

inpart(a)consists offat?

(d)

The body volume

calculatedinpart(a)includesvolumesoccupiedbygasinthedigestivetract, sinuses, andlungs.

The sum

of thefirst two volumesisroughly 100

mL

and thevolume of the

5Ifyoucan'tworkoutthe proof, takethe given formulaas validandproceedto thenextpart.

Problems

₆₇ lungsisroughly_1.2liters.

Hie

massof the gasisnegligible.

Use

thisinformationtoimproveyour estimate of x(^.

3.12.

Aqueous

solutionsof the

amino

acid L-isoleucine(He) areprepared byputting 100.0gramsofpure water intoeachofsix flasks andaddingdifferent preciselyweighedquantities of

He

to each flask

The

densitiesof thesolutions_at50.0

±

0.05°Carethen

measured

withaprecisiondensitometer, with the following results:

r (gIle/100 g

H

2

0)

0.0000 0.8821 1.7683 2.6412 3.4093 4.2064 p(gsolution/cm³

) 0.98803 0.98984 0.99148 0.99297 0.99439 0.99580

(a) Plot a calibrationcurve showingthe massratio,r,asa function of solution density, _p,andfita straightlineto the datatoobtainan equationofthe

form

r

=

_ap

+

b.

(b)

The

volumetric flowrate ofan aqueous

He

solution at a temperature of ^50°

C

^{is 150} L/h.

The

densityof asample ofthe streamis measuredat50°

C and

found to be0.9940g/cm³.

Use

the calibration equationtoestimatethemassflowrate of

He

inthestream (kg Ile/h).

(c) Ithasjustbeendiscoveredthatthethermocouple usedtomeasurethestreamtemperaturewas poorlycalibratedandthetemperaturewasactually47°^C.

Would

thelie

mass

flowrate calculated inpart (b) be too high or too low? State any assumption you

make

and brieflyexplain your reasoning.

3.13. Beforearotametercanbe usedtomeasure an

unknown

flowrate,a calibrationcurveofflow rate versusrotameterreadingmustbe prepared.

A

calibrationtechniquefor liquidsisillustratedbelow.

A

^flowrateissetbyadjusting the

pump

speed;the rotameter readingisrecorded,andthe effluent

from

therotameteriscollectedinagraduatedcylinder foratimedinterval.

The

procedureiscarried out twiceforeachofseveral

pump

settings.

ROTAMETER

VARIABLE SPEED

PUMP

GRADUATED CYLINDER

Rotameter _Collection

Volume Reading Time

(min) Collected(cm³₎

2 1 297

2 ₁ 301

4 ₁ 454

4 1 448

6 0.5 300

6 0.5 298

8 _0.5 371

8 _0.5 377

10 0.5 440

10 0.5 453

(a)

Assuming

the liquidiswater at25°C,

draw

a calibration curve ofmass flow rate,

m

^(kg/min),

versus rotameterreading, _R, and use it to estimate the mass flow rate of a water stream for whichtherotameterreadingis5.3.

(b)

The mean

differencebetweenduplicates,

W

h provides anestimateof thestandard deviation of a singlemeasurement, which wasgiventhesymbols_x

on

p. 18 ofChapter2:

=

_0.8862D,

Moreover,confidencelimitson measuredvalues canbeestimatedtoa

good

approximationus- ingthe

mean

differencebetweenduplicates.For example,ifa single

measurement

ofYyieldsa value /"measured, then there is a

95%

probability that the true value of

Y

falls within the

95%

confidence _limits (y

m

e asured

~

1.74D,) and (Y_meisund

+

1.74D,).⁶ For a

measured

flow rate of 610g/min, estimate the

95%

confidencelimitson the trueflowrate.

6W.Volk,AppliedStatisticsforEngineers,McGraw-Hill,

New

York,_pp.113-115,1958.

3.14.

How many

ofthe following arefoundin15.0

kmol

ofbenzene (C6

H

6)? (a)kg

C

6

H

6^;(b)

mol C

6

H

6;

(c)lb-mole

C

6

H

6; (d)

mol

(g-atom) C;(e)

mol

H; (f) g C;(g) gH;(h)molecules of

C

6

H

6^. 3.15. Liquid tolueneisflowingthroughapipeata rateof175

m

³/h.

(a)

What

isthe massflow rateofthisstreaminkg/min?

(b)

What

isthe molarflowrateinmol/s?

(c) In fact,the answer topart (a) isonlyan approximation thatis almostcertain tobe slightly in error.

What

didyou havetoassumetoobtain theanswer?

3.16.

A

mixture ofmethanol and methylacetatecontains 15.0

wt%

methanol.

(a) Usinga singledimensional equation,determinetheg-molesofmethanolin200.0kgof themix- ture.

(b)

The

flowrateofmethylacetatein themixtureistobe 100.0 lb-mole/h.

What must

themixture flow ratebein lbm^/h?

3.17.

The

feed toan

ammonia

synthesis reactor contains25

mole%

nitrogenand thebalance hydrogen.

The

flow rate ofthe stream is3000kg/h. Calculate the rate of flow of nitrogeninto the reactor in Student k_§/h- (Suggestion:Firstcalculate theaverage molecular weight ofthe mixture.)

orkbook

3>18>

A

suspension of calcium carbonate particles in waterflows through a pipe.

Your

^assignment is to

determineboththeflowrateandthecomposition ofthis slurry.

You

proceedtocollectthestreamin a graduatedcylinder for 1.00min;youthen weighthe cylinder, evaporate thecollected water,and reweigh the cylinder.

The

followingresultsareobtained:

Mass

of

empty

cylinder: 65.0 g

Mass

ofcylinder+collectedslurry: 565 _g

Volume

collected: 455

mL

Mass

of cylinderafterevaporation: 215_g Calculate

(a) thevolumetricflow rate and massflow rate of thesuspension.

(b) the densityofthesuspension.

(c) themassfractionofCaCC>3 inthe suspension.

A

^{mixtureis10.0}

mole%

ethyl alcohol, 75.0

mole%

ethylacetate(C4

H

8

0

2),and15.0

mole%

acetic acid. Calculate themassfractions ofeachcompound.

What

isthe average molecular weight of the mixture?

What

wouldbethe mass(kg)ofa samplecontaining25.0

kmol

ofethylacetate?

Certainsolidsubstances,

known

ashydratedcompounds, havewell-definedmolecularratiosof water to

some

otherspecies,whichoftenisasalt.Forexample, calciumsulfatedihydrate

(commonly known

asgypsum,

CaS0

4^•

2H

20), has² molesof waterpermole ofcalcium sulfate; alternatively,it

may

besaid that1 moleof

gypsum

consistsof1moleofcalciumsulfateand2molesofwater.

The

water insuch substancesiscalledwater ofhydration.

(More

information about hydratedsalts isgiven in

Chapter6.)

Solid

gypsum

isformed inacrystallizerandleaves that unit as a slurry (asuspension ofsolid particlesinaliquid)ofsolid

gypsum

particlessuspendedinan aqueous

CaS0

₄solution.

The

slurry flows from the crystallizerto a filter inwhich the particles are collected as afilter cake.

The

filter cake,whichis95.0

wt%

solid

gypsum

andtheremainder

CaS0

4solution,isfedto adryerinwhich

all water(including the water of hydrationinthe crystals)isdriven off to yield anhydrous(water- free)

CaS0

4asproduct.

A

flowchartandrelevantprocess dataaregiven below.

Solidscontent ofslurryleavingcrystallizer:0.35kg

CaS0

4 ^•

2H

₂

0/L

slurry

CaS0

₄content ofslurryliquid:0.209g

CaSO

4/100g

H

2

0

Specificgravities:

CaSQ

4^•

2H

2Q(s),2.32;liquidsolutions, 1.05 3.19.

Equipment Encyclopedia

crystallizer, filter,dryer

CaS0₄(aq)

Water vapor CRYSTALLIZER

Slurry

FILTER

Filtercake DRYER

Incompletelylabeled flowchart

Filtrate CaSOi