SUMMARY

ABOUT CHEMICAL PROCESSES

4.10 SUMMARY

valuesandvaluescalculatedfromthem.Moreover, any measured^value(e.g.,aninputorout- putstream volumetric flow^rate,the

mole

fractionof

MEK

^{inthefeed orvapor}product stream, anystreamtemperatureor pressure)issubject to errorsduetoa faultyinstrument(e.g.,amal- functioning or poorly^calibrated flowmeter or gaschromatograph)or

random

datascatter.

• Impuritiesin thefeed.

The

designcalculationswerebasedon an assumptionthat the feed contains only

MEK

^{vaporandnitrogen.} Impurities presentinthe feedcouldreactwith the

MEK,

orthey couldcondense and^affectthe vapor-liquid equilibriumdistributionof

MEK

intheproducts.

• Incorrect assumption of^{steady state.} Closure should be expected only after the system reaches steady state, so ^that input

=

output. In the experimental run, steady state was declared

when

^the operator could no longer see changes in the outlet stream rotameter readings. It is possible that the flow rates were still changing but the rotameter was not sensitive

enough

to

show

the changes.It isalso possible that

MEK

wasstillaccumulatingin thesystem

—

^{forexample,byadsorbing}

^on

the containerwalls

—

^and

^{much more}

^timewould

berequiredfor thebuilduptobecomplete.

• Incorrect assumption that

MEK

^is not reactive. If

MEK

^{undergoes a reaction in the}

system—

decomposition,forexample, or reaction withsomething onthe reactor

wall—

then input

=

output

+

consumption.

The

output

would

then necessarily be less than the input and ^thebalance

would

notclose.

• Errors dueto approximations inthe experimental data^analysis. Severalpotential errors were introduced

when

the measured volumetric flow rates were converted to molar flow rates. Volumetricgasflowrateswere convertedusing^{the ideal}gas equation of^state,which

isapproximate,andthevolumetricliquid flowratewasconverted usinga tabulated^density that

may

nothavebeen measured atthesystem temperature.^Also, thefact thata physical property value hasbeenpublishedis

no

^{guaranteethat itiscorrect.}

• Approximationsinthedesignanalysis. Liketheidealgasequation ofstate,Raoult's lawis

an approximation^that

may

^{beexcellentorseriouslyinerror,depending}

^on

^theexperimental processconditions.

There areotherpossibilities,butyouget the idea.

The

point^is that nomatter

how

^{carefully you}

designa process,

you

cannot predict exactlywhattherealprocesswilldo. Approximations and^as- sumptions must be

made

for every process design; closures on realprocess material balances ^are neverexactly

100%;

nothingcan be measured ^with complete accuracy; and everyone sometimes

makes

^mistakes.

Experienceddesign engineers

know

these thingsand accountfor

them

with overdesignfactors.If

theycalculate thattheyneeda 2500-liter reactor,theymightordera 3000-liter or 3500-liter reactor to

make

^{sure theyhave}

^enough

reactor capacityto

meet

bothcurrentandanticipatedproductdemands.

The more

uncertaintiesinthe design or the projected productdemand,the greater the overdesign.

A

large part ofwhatengineers

do

involvesreducing theuncertaintiesandthuslowering^therequired overdesign,^resultinginmajorreductionsinequipmentpurchaseand maintenance^costs.

4.10

SUMMARY

Every

chemicalprocess analysisinvolves writing

and

solving materialbalances to account^for

allprocess speciesinfeed

and

^{productstreams.Thischapteroutlines}

and

illustratesa systematic

approach

to material balance calculations.

The

procedure is to

draw and

^label a flowchart,

perform

adegree-of-freedomanalysis to verify that

enough

equations can be^{writtentosolve} for all

unknown

^{processvariables,}

^and

^write

^and

solve the equations.

•

The

generalbalance equation^is

input

+

generation

-

output

— consumption =

accumulation

A

differential balance applies to an instant of time

and

each

term

is a rate (mass/time or moles/time).

An

^{integralbalanceappliestoatimeinterval}

^and

^each

^term

^isan

^amount

^(mass

or moles). Balances

may be

appliedto total mass, individualspecies, or energy. (They

may

alsobe applied to

momentum,

but

we

willnot^consider

momentum

^balancesinthis text.)

• For a differential balance

on

a continuous process (material flows in

and

out throughout the process)atsteady-state (noprocessvariableschange withtime),theaccumulation term

inthebalance (the rate ofbuildupor depletion of thebalanced species)equalszero.For an integralbalance

on

abatch process (nomaterial flowsinor outduringthe process), the input

and

output terms equal zero

and

accumulation

=

initialinput

-

_{finaloutput. Inbothcases,}

thebalancesimplifiesto

input

+

generation

=

output

+ consumption

Ifthe balanceis

on

total

mass

or

on

a nonreactivespecies,theequationsimplifiesfurtherto input

=

output

•

A

process stream

on

a flowchart is completely labeled ifvalues or variable

names

are as- signed to

one

of thefollowingsetsofstream variables: (a) total

mass

flowrate ortotal

mass and component mass

fractions; (b)

mass

flowratesormassesofeach stream

component;

(c) total

molar

flow rateortotal

moles and component mole

fractions;

and

(d)

molar

flowrates ormolesofeach

stream component.

_Ifatotal

amount

orflow^rateor

one

or

more _component

fractionsare

known

for astream, use(a) or(c) toincorporatethe

known

valuesinto the label- ing. If neither the total

nor any

fractions areknown, using (b) or(d)

(component amounts

or flowrates) often leads toeasieralgebra. Volumetric_quantities should belabeled onlyifthey areeithergiven orrequestedinthe

problem

statement.

A

^flowchartiscompletelylabeledif

every streamiscompletelylabeled.

•

A

^basisof calculation for aprocessisan

amount

or flowrateof

one

of the process streams. If

two

or

more

stream flowratesor

amounts

are giveninthe

problem

statement,theyconstitute the basis of calculation. If

one

is given, it

may

be

assumed

as a basis but it

may

also be convenientto

assume

anotherbasis

and

thenscale the flowchart to thespecified value.If

no

flow rates or

amounts

are given,

assume one

as a basis, preferably

an amount

of a stream with

known

composition.

•

To perform

a degree-of-freedom _analysis

on

a single-unit nonreactive _process, count un-

known

variables

on

the flowchart, then subtract independent relations

among

them.

The

difference,

which

_equalsthe

number

ofdegrees of

freedom

forthe process,

must

equal zero forauniquesolutionofthe

problem

tobedeterminable. Relations include material balances (as

many

asthere are

independent

speciesinthe feed

and

productstreams), processspecifi- cations,density relations

between

labeledmasses

and

volumes,

and

physical constraints(e.g.,

the

sum

of the

component _mass

_or

_mole

_{fractionsof astream}

must add up

to_1.)

•

To perform

a degree-of-freedomanalysis

on

a multiple-unit process,

perform

separate_anal- yses

on

the overall process,

each

processunit,each stream mixingorstreamsplitting point, and, ifnecessary,

on

combinations of processunits.

When you

finda system with zero de- grees offreedom,

assume

that

you

cansolveforthe

unknown

variablesinthe feed

and

out- put streams forthatsystem;then, considering those variables as

known,

try to findanother system with zerodegrees of freedom. Thisprocedurehelps

you

to findan efficientsolution procedure before

you undertake

time-consumingcalculations.

•

Once you have

written thesystemequationsforaprocess,

you may

solve

them

either

manu-

allyor using

an

equation-solving

computer

program._If

you

solvesystemequations manually, write

them

in

an

orderthatminimizesthe

number

that

must

be solvedsimultaneously, starting with equationsthatonlyinvolve

one unknown

variable.

• Recycleisa

common

feature ofchemical processes.Its

most common

useis tosend

unused raw

materials

emerging from

a processunitbacktotheunit.Overallsystembalances are usu- ally (butnotalways) convenient startingpointsforanalyzing process withrecycle.

A

purge stream is

withdrawn from

a process

when

aspecies enters inthe process feed

and

is

com-

pletely recycled. Ifthis species

were

not

removed

in the purge,it

would keep

accumulating inthe processsystem

and

eventually leadtoshutdown.

•

The

limitingreactantina reactive processisthe one that

would

be completely

consumed

if

the reaction

proceeded

tocompletion.Allother reactants

must

eitherbefedinstoichiometric

Problems

₁₅₅

Interactive Tuturials

#3

Questions with Immediate Feedback

E-ZSolve Solves complicated equationsquickly

PROBLEMS

4.1.

4.2.

proportionto the limiting reactant (the feed rates arein theratioof the stoichiometric co- efficients) or in excess of the limiting reactant (in greater than stoichiometric proportion toit).

•

The

theoreticalrequirementforanexcess reactantisthe

amount

requiredtoreactcompletely with thelimiting reactant.

The

percentage excessof the reactantis

amount

fed

- amount

theoreticallyrequired

%

^excess

⁼

^{: :}

— — ^-

amount

theoretically required

The

percentageexcess

depends

only

on

the feed rates of the excess

and

limitingreactants

and on

theirstoichiometriccoefficients;it does not

depend on how much

actuallyreacts or

on

anythingelse that

happens

inthereactor.

•

The

fractional conversion of a reactant is the ratio of

amount

reacted to

amount

fed.

The

fractionalconversionsofdifferentreactants are generallydifferentunlessthe reactants are fedinstoichiometric proportion.

•

The

extent of reaction, £ ^(or£ for acontinuousprocess), isaspecies-independent quantity thatsatisfiestheequation

rij

=

n_iQ

+

or

=

h_i0

+

v,£

where

n

i0 (n_{i0 )} is the

number

of

moles

(molarflow rate) of speciesi in the feed to there- actor,rii (hi) ^isthe

number

of

moles

(molarflowrate) of speciesi inthestreamleaving the reactor,

and

v, is the stoichiometriccoefficientof speciesi (negative for reactants, positive forproducts,

and

zero for nonreactive species).

The

units of £ (£) are the

same

asthose of n ^{(h). If}

you know

the inlet

and

outlet

amounts

or flow rates ofany reactive species,

you

can determine_{£ or £}

by

applyingthis equationto thatspecies.

You may

thensubstitutethe calculatedvalueinto theequationsfortheotherspeciesin thestreamleaving the reactorto determinethe

amounts

or flowrates ofthosespecies.

•

You may

analyze reactive processes using (a)molecularspecies balances (the only

method

usedfornonreactiveprocesses),(b)atomicspeciesbalances,or(c)extentsof^reaction. Molec- ular speciesbalances

on

reactiveprocesses are often

cumbersome:

they

must

include gener- ation

and consumption

terms for eachspecies,

and one

degree of

freedom must

be

added

foreach independentreaction.

Atomic

speciesbalanceshave thesimple

form

input

=

_out-

put and

are usually

more

straightforward thaneitherof the other

two

methods. Extentsof reaction areparticularlyconvenient forreaction equilibriumcalculations.

•

Combustion

^isarapid reaction

between

afuel

and

oxygen.

The

carboninthefuelisoxidized to

CO2

(complete combustion) or

CO

(partial combustion)

and

the

hydrogen

in the fuel is oxidized to water.

Other

species in the fuel like sulfur

and

nitrogen

may

be partiallyor completely converted to their oxides.

Combustion

reactions are carried out commercially eithertogenerateheat orto

consume

wasteproducts.

Note: This

would

bea

good

timeto

work

throughInteractive Tutorial#3.In the

problems

that follow,

you

canuse

E-Z

Solvetosolve setsofequationsquickly.

Water

enters a2.00-m³ tankata rateof6.00 kg/sandiswithdrawnata rateof3.00kg/s.

The

tankis initiallyhalffull.

(a) Isthisprocess continuous,batch,or semibatch?Isittransientorsteadystate?

(b) Write a mass balance for the process (see

Example

4.2-1). Identify the terms ofthe general balanceequation (Equation4.2-1) presentin yourequationand state the reason foromitting anyterms.

(c)

How

^longwillthetank taketooverflow?

A

liquid-phase chemicalreaction

A ^—

^*

B

takes place ina well-stirred tank.

The

concentration of

A

^{inthe feedis}

C A

o (mol/m³), andthat in thetankand outletstreamis

C

A (mol/m³).Neither con- centration varieswithtime.

The volume

ofthetank contentsis

V(m

³) andthe volumetric flowrate of the ^inletand outlet streams isv

(m

³/s).

The

reaction rate (the rate atwhich

A

is

consumed

by reactioninthetank) isgivenby^{theexpression}

r(mol

A

consumed/s)

= kVC

_A

wherek isa constant.

156

Chapter

4

Fundamentals

ofMaterial Balances u(m³/s)

Equipment Encyclopedia evaporator

4.3.

4.4.

(a) Is thisprocess continuous,batch,orsemibatch?Isittransientorsteady-state?

(b)

What

would youexpect thereactantconcentration

C

A ^to equal if k

=

_{0 (no reaction)?}

_What

shoulditapproachifk -* °° (infinitelyrapid reaction)?

(c) Write a differential balance

on

A, stating which terms in the general balance equation (accumulation

=

input

+

^generation

-

output

-

consumption)

you

discarded and

why

you discarded them.

Use

the balance toderive the following relation betweenthe inlet and outlet reactantconcentrations:

C = ^

A0 A

1

+ kVjv

Verifythat thisrelation predicts theresults inpart(b).

A

liquid mixture ofbenzene andtoluene contains55.0% benzene by mass.

The

mixture is to be partiallyevaporated to yield a vapor containing 85.0% benzene and a residual liquid containing 10.6% benzene bymass.

(a) Supposetheprocessistobecarriedout continuouslyandatsteadystate,withafeedrate of 100.0 kg/h ofthe

55%

mixture.Let

m

v(kg/h)andmi(kg/h)bethemassflowrates of thevapor andliquid productstreams, respectively.

Draw

andlabela processflowchart,thenwriteandsolvebalances

on

total mass and

on

benzene to determine the expected values of

m

v and

m

_h For each bal- ance, statewhichterms of the general balance equation (accumulation

=

_input

+

generation

-

output

-

consumption)

you

discardedand

why

youdiscarded them. (See

Example

4.2-2.) (b) Next, supposetheprocessistobecarriedoutina closedcontainerthatinitiallycontains100.0kg

ofthe liquid mixture.Let

m

v(kg)and_{m,(kg) be}themasses of thefinalvapor and liquidphases, respectively.

Draw

andlabela processflowchart,thenwriteandsolve integralbalances

on

total

'

mass and on benzenetodetermine

m

vand

m

x.For eachbalance,statewhichterms ofthegeneral balance equation (accumulation

=

input

+

generation

-

_output

-

consumption)

you

discarded and

why

youdiscarded them.

(c) Returning tothe continuous process, suppose the evaporator is built and started up and the productstream flowratesandcompositionsaremeasured.

The measured

percentage ofbenzene

in thevaporstreamis

85%

andtheproduct streamflow rateshavethevaluescalculatedinpart (a),buttheliquidproductstreamisfoundtocontain

7%

benzeneinsteadof10.6%.

One

^possible

explanation is that a mistake was

made

in the measurement. Give at least five others. [Think aboutassumptionsyou

made

inobtainingthe solutionofpart(a).]

Draw

and label the given streams and derive expressions for the indicated quantities in terms of labeledvariables.

The

solution ofpart (a)isgivenasanillustration.

(a)

A

continuous stream contains40.0

mole%

benzene andthebalance toluene.Writeexpressions forthe molar and

mass

flowratesof benzene,_{n B}(mol

C

6

H

6/s) and /nB(kg

C

6

H

6 /s), interms of thetotalmolarflowrateof the stream,«(mol/s).

Solution

n(mol/s)

—

^,

—

0.400

mol C

6

H

6/mol 0.600

mol C

7

H

8/mol n_B

=

_0.400«(mol

C

6

H

6 /s)

m

_B ^0.400n(mol

C

6

H

_{6 )}

78.1gC

6

H

6

mol

31.2n(g

C

₆

H

6/s)

Problems

157

(c)

(d)

(b)

The

feed toa batch process containsequimolarquantitiesof nitrogenand methane. Write an expressionforthe kilograms of nitrogeninterms ofthe totalmoles n(mol) ofthismixture.

A

streamcontaining ethane, propane, and butane has a mass flow rate of 100.0 g/s. Write an expressionforthemolarflowrateofethane, hB (lb-mole

C

2

H

6/h), intermsof the massfraction ofthisspecies,x_E.

A

continuousstream of

humid

aircontains watervapor and dryair, the lattercontaining ap- proximately 21

mole% 0

2 and

79% N

2^. Write expressions for the molar flowrate of

0

2 and forthemolefractionsof

H

2

0

^and

0

2 inthe gasinterms of/^(lb-mole

H

20/s) and ri

2(lb-mole dryair/s).

(e)

The

product from a batch reactor contains

NO, N0

2^, and

N

₂

0

4^.

The mole

fraction of

NO

^is

0.400.Writeanexpressionforthegram-molesof

N

2

0

4interms of«(molmixture)and_vn0 (mol

N0

2/mol).

2

5. (a)

Draw

aflowchartforthecatalyticdehydrogenationofpropanefromthedescription ofthispro- cess thatbegins Section4.3a.Labelallfeed,product,andconnecting streams betweenunits.

(b) Writeclearstatements ofthe overall objective ofthe process andthe functions ofeach of^the processunits(thepreheater, thereactor,theabsorptionandstrippingtowers,andthedistillation column).

6.

A

distillation

column

isaprocessunitinwhichafeed mixtureisseparatedbymultiplepartialvapor- izationsand condensations to

form

two or

more

productstreams.

The

overhead product streamis

richinthemostvolatile

components

ofthefeed mixture (theonesthatvaporizemostreadily),and thebottomproductstreamisrichintheleast volatilecomponents.

The

following flowchart showsa distillation

column

with twofeedstreams and threeproduct streams:

m

₃(kgA/h)

Wkg/h) 0.03kg B/kg 0.97kg C/kg

5300kg/h -r2(kg A/kg)

y2 (kgB/kg)

1200kg/h 0.70kg A/kg v4 (kgB/kg) z 4 (kg C/kg)

m

₅(kg/h) 0.60 kg B/kg 0.40 kgC/kg

(a)

How many

independentmaterial balances

may

bewritten forthissystem?

(b)

How many

of the

unknown

flowratesand/or

mole

fractionsmustbespecifiedbeforetheothers

may

becalculated? (See

Example

^4.3-4. Also,

remember

whatyou

know

about the

component

molefractions ofamixture

—

^forexample, therelationshipbetweenx₂ and_y2.)Brieflyexplain your answer.

(c) Supposevalues are givenforih\ andx2^.Giveaseriesofequations, eachinvolvingonly a single

unknown,

for theremainingvariables. Circle thevariable forwhich you wouldsolve. (Oncea variable hasbeencalculated inoneof these equations,it

may

appearin subsequent equations without beingcountedasanunknown.)

7. Liquid^extractionisanoperation usedtoseparatethecomponentsofa liquidmixture oftwoor

more

species.In thesimplestcase,themixture containstwo components:asolute(A)anda liquid solvent (B).

The

mixtureiscontactedinanagitated vesselwithasecondliquidsolvent(C)thathastwo key properties:

A

^dissolvesinit,and

B

^isimmiscible or nearly immiscible withit.(Forexample,

B may

be water,

C

^ahydrocarbonoil,and

A

a species that dissolvesinbothwaterandoil.)

Some

^{of the}

A

transfers from

B

to C,

and

then the B-richphase(the raffinate) andthe C-rich phase (the extract) separatefrom eachotherinasettlingtank.Ifthe raffinateisthen contacted withfresh

C

^inanother

4.8.

4.9.

4.10.

Student Workbook

stage,

more A

willbetransferredfromit.Thisprocess can be repeateduntil essentiallyallof the

A

hasbeenextractedfromthe B.

Shown

belowisa flowchart of a processinwhichacetic acid(A)isextracted

from

amixtureof aceticacidandwater (B) into 1-hexanol(C),a liquidimmiscible withwater.

m

_{c (g}C₆H₁₃OH/min)

m

_E(g/min)

400g/min

0.115gCH₃C00H/g 0.885_gH₂0/g

0.096 gCH₃C00H/g 0.904gC_6Hi30H/g

m

_R(g/min)

0.005_gCH₃C00H/g 0.995 gH₂0/g

(a)

What

is the

maximum number

ofindependent material balances that can be written forthis process?

(b) Calculate

m

_c,

m

E, and

m

R,usingthe given mixture feed rate as abasisandwriting balancesin an order suchthatyou never have anequationthat involves

more

thanone

unknown

variable.

(c) Calculate the difference between the

amount

of acetic acidin thefeed mixtureand thatinthe

0.5%

mixture,and

show

thatitequals the

amount

that leavesinthe

9.6%

mixture.

(d) Aceticacidisrelatively difficulttoseparate completelyfromwaterbydistillation(seeProblem

4.6)andrelativelyeasytoseparatefrom hexanol bydistillation.Sketch a flowchart ofatwo-unit processthatmight be usedtorecover nearly pureacetic acidfromanaceticacid-water_mixture.

Eggsare sortedintotwosizes(large

and

extra large)attheCheerfulChickenDairy.Unfortunately, business has notbeen goodlately,andsincetheCheerful Chicken's40-year-old egg-sortingmachine

finallygave

up

theghosttherehave been

no

fundsavailable to replaceit. Instead,

Old

Fred,oneof thefirm'ssharper-eyed employees, hasbeen equippedwitha"Large"rubberstampinhisright

hand and

an "X-large"stamp in hisleftand assigned tostamp each egg withthe appropriate label asit

goes by on the conveyor belt.

Down

the line, another employee putsthe eggs into either of two hoppers, each egg according toitsstamp.

The

systemworksreasonablywell,allthingsconsidered, except that Old Fred has a heavy

hand

and

on

the average breaks

30%

ofthe 120 eggs that pass by

him

each minute.

At

the

same

time, acheck of the "X-large" stream reveals a flow rate of70 eggs/min, ofwhich25 eggs/min are broken.

(a)

Draw

andlabela flowchartforthisprocess.

(b) Write andsolvebalancesabouttheeggsorteron totaleggsand brokeneggs.

(c)

How many

"large"eggs leavethe planteachminute,and whatfractionof

them

arebroken?

(d) Is

Old

Fredright-orleft-handed?

Strawberries contain about 15

wt%

solids and 85

wt%

water.

To make

strawberry jam, crushed strawberriesandsugarare

mixed

ina 45:55massratio,andthemixtureisheatedtoevaporate water untiltheresidue containsone-thirdwater bymass.

(a)

Draw

andlabelaflowchart ofthisprocess.

(b)

Do

thedegree-of-freedomanalysisand

show

that thesystem has zero degrees offreedom(i.e.,

the

number

of

unknown

processvariablesequals the

number

ofequationsrelatingthem).Ifyou havetoo

many

unknowns,think about

what

you might haveforgottentodo.

(c) Calculate

how many

poundsof strawberriesareneededto

make

a

pound

ofjam.

Three hundredgallonsof a mixture containing 75.0

wt%

ethanol (ethylalcohol) and

25%

water (mixturespecificgravity

=

0.877)

and

a quantity ofa 40.0

wt%

ethanol-60% watermixture

(SG =

0.952) areblendedtoproduce amixture containing60.0

wt%

ethanol.

The

objectofthisproblemis todetermine V^,therequired

volume

ofthe

40%

mixture.

(a)

Draw

andlabelaflowchart of themixingprocessand

do

thedegree-of-freedom analysis.

(b) Calculate

V m

^.

Ifthe percentageoffuel ina fuel-airmixturefallsbelowa certainvaluecalled thelowerflammability limit(LFL), themixture cannotbe ignited.Forexample,the

LFL

ofpropanein air is2.05

mole%

C,H

8. Ifthe percentageofpropane ina propane-air mixtureisgreaterthan 2.05

mole%,

the gas mixture canigniteif it is exposedtoa flame orspark; ifthepercentageislower than the

LFL,

the mixturewillnotignite. (Thereisalsoanupperflammabilitylimit,whichforpropanein airis11.4%.)