MASS AND VOLUME

The

density of asubstanceisthe

mass

perunit

volume

of thesubstance

(kg/m

³,

g/cm

³,lb

m

^/ft3,

etc.)

The

specific

volume

of a substanceisthe

volume

occupied

by

a unit

mass

of the substance;

it isthe inverse ofdensity. Densities ofpuresolids

and

liquidsare essentiallyindependent of pressure

and

vary relatively slightlywith temperature.

The

temperature variation

may

be in either^direction: thedensity of^liquidwater, forexample,increases

from

0.999868 g/cm³ at

0°C

to 1.00000

g/cm

³ at3.98°C,

and

then decreases ^to0.95838

g/cm

³ at 100°C. Densities of

many

pure

compounds,

^solutions,

and

mixtures

may be found

instandardreferences (suchas Perry's

Chemical

Engineers'

Handbook}

pp. 2-7 through2-47

and

^2-91 through 2-120).

Methods

of estimatingdensitiesof gases

and

mixturesofliquidsare givenin

Chapter

5 ofthisbook.

The

density of a substance can be

used

as a conversionfactorto relate the

mass and

the

volume

of a quantity of the substance.For example,the density ofcarbontetrachlorideis1.595

g/cm

³;the

mass

of 20.0

cm

³ of

CC1

₄is therefore

20.0

cm

³ _{1.595 g}

cm

³

=

_{31.9 g}

and

^the

^volume

^{of 6.20lb}

m

^of

^CCI4

^is

6.20lbm 454 g ¹

cm

³

Hbm

^1.595 g

The

specific gravityof a substance^istheratioofthe density

p

^{of thesubstanceto}the density

p

refofa referencesubstanceataspecificcondition:

SG ⁼ _p/p

_ref

^(3.M)

'R.H. Perryand D. W.Green.Eds.,Perry'sChemicalEngineers'Handbook,7th Edition, McGraw-Hill,

New

York,1997.

The

reference

most commonly

usedfor solids

and

liquids iswaterat4.0°C,

which

has the followingdensity:

PH

20(i)(4°C)

=

1.000g/cm³

=

_1000.

_kg/m

³

=

62.43 lb

m

^/ft3

(3.1-2)

Note

thatthedensity of aliquidorsolid in

g/cm

³ isnumerically equaltothespecificgravityof thatsubstance.

The

notation

SG ⁼

0.6 20°

signifiesthatthespecificgravityof asubstanceat

20°C

with referencetowaterat

4°C

is0.6.

If

you

are given the specificgravity

of

^a substance, multiply itby the referencedensity in

any

^unitstoget the density

of

^thesubstancein the

same

units. For example, ifthespecificgrav- ity of a liquid is 2.00, its density is 2.00

x

10³

kg/m

^J or 2.00

g/cm

³ or 125 lb

m

^{/ft3. Specific}

gravitiesof selectedliquids

and

solidsaregiveninTableB.l.

Note: Special density units called degrees

Baume

(°Be), degrees

API

(°API),

and

degrees Twaddell (°Tw) are occasionally used, particularly in the

petroleum

industry. Definitions of

and

conversionfactors fortheseunitsaregiven

on

p.1-20 ofPerry^'s

Chemical

Engineers'

Hand-

book.

TEST

¹

YOURSELF

²

(Answers,

_p.

₆₅₅₎

What

are theunitsofspecificgravity?

A

^liquid has aspecific gravity of0.50.

What

is its density in

g/cm

³?

What

is its specific

volume

in

cm

³/g?

What

isitsdensityin lb

m

^/ft3?

What

isthe

mass

of 3.0

cm

³ofthisliquid?

What volume

is occupied

by

18g?

Ifsubstance

A and

substance

B

each

have

a density of1.34

g/cm

³,

must

3

cm

³ of

A

^have

the

same mass

as3

cm

³ of

B?

Ifsubstance

A and

substance

B

each

have

aspecificgravityof1.34,

must

3

cm

³ of

A

have the

same mass

as3

cm

³ of

B? Why

not?

Freezing a sealed fullbottle ofwater leads to a

broken

bottle

and

freezing a sealedfull flexible-walledcontainer ofrt-butylalcohol leadstoacontainer with

concave

walls.

What

can

you

conclude aboutthe_densitiesof thesolid

and

liquidformsof these

two

substances?

Does

the density ofliquid

mercury

increase or decrease with increasing temperature?

Justifyyouranswerusinga

thermometer

asan illustration.

EXAMPLE

3.1-1,

Mass, Volume, and Density

Calculate thedensityofmercuryin lb

m

^/ft3froma tabulatedspecific gravity,andcalculatethevolume

inft3

occupied by 215 kg of mercury.

SOLUTION

TableB. 1 liststhespecificgravityofmercuryat20°Cas13.546.Therefore,

PHg

=

(13.546) (62.43

^

^{845.7 lb,ft3}

215 kg llb m ^1ft3 0.454kg 845.7 lbm

0.560ft3

3.2

Flow Rate

45

As

statedearlier, temperature

and

pressure

do

not

have

largeinfluences

on

thedensities ofsolids

and

liquids. Nevertheless, the fact that

mercury

in a

thermometer

rises or falls with changing temperature

shows

that the effect of temperature

on

liquid density is measurable.

Coefficientsoflinear

and

cubic(volume) thermal expansion of selectedliquids

and

solidsare givenasempiricalpolynomialfunctions oftemperature

on

pp. 2-128 ^to2-131 ofPerry's

Chem-

icalEngineers'

Handbook.

For example,the

Handbook

givesthe

dependence

of the

volume

of

mercury on

temperatureas

V(T) = V

₀^(l

+

0.18182

X

10_3

r +

0.0078

X

^10~67/2

) (3.1-3)

where V(T)

isthe

volume

of a given

mass

of

mercury

attemperature7/(°C)

and V

₀isthe

volume

of the

same mass

of

mercury

at0°C.

EXAMPLR3.1-2\

Effect

of Temperature on Liquid Density

In

Example

^3.1-1,215kgofmercury was foundtooccupy0.560ft3

at20°C. (1)

What volume

would the mercuryoccupyat100°C?₍₂₎Supposethemercuryiscontainedina cylinderhavingadiameter of0.25 in.

What

changeinheightwould be observedas themercuryisheatedfrom

20°C

to100°C?

SOLUTION

^1.

^From

^{Equation3.1-3}

V(100°C)

= V

_{0 [l}

+

0.18182

X

10^_3(100)

+

0.0078

X

^10~6₍₁₀₀₎²

]

and

V(20°C)

=

0.560ft

3

= V

0[l

+

0.18182

X

^10~3(20)

+

0.0078

X

10^_6(20)²]

Solvingfor

V

0

from

thesecond equationandsubstitutingitintothe firstyields V(100°C)

=

_0.568ft³

2.

The volume

of the mercury equals ttD²

h/

4,

where D

^is the cylinder diameter and

H

^{is its}

height.Since

D

^isconstant,

tf(100°C)

-

_tf(20°C)

=

^y(ioo°C)

-

_y(20°c)

ttD²/4

D

= _(0.25/12)ft

23.5ft

3.2

FLOW RATE

3.2a Mass and Volumetric Flow Rate

Most

processes involve the

movement

of material

from one

point toanother

— ^sometimes

^be-

tween

processunits,

sometimes between

aproductionfacility

and

a transportation depot.

The

rate at

which

a materialistransportedthrougha processlineisthe flow rate ofthatmaterial.

The

flowrate of a processstream

may

be expressedasa

mass

flowrate (mass/time) or as a volumetricflow rate (volume/time).

Suppose

a fluid (gas or liquid) flows in the cylindrical pipe

shown

below,

where

theshaded area represents a section perpendicular to the direction

m(kgfluid/s)

VXm³fluid/s)

46

Chapter

3 Processes

and

Process Variables

of flow. Ifthe

mass

flow rate ofthe fluid is m(kg/s)², then every second

m

kilograms of the fluid pass throughthe cross section. If thevolumetricflow rateof the fluid atthe given cross sectionis

V(m

³/s),thenevery second

V

^cubicmetersofthefluidpassthroughthe crosssection.

However,

the

mass m and

the

volume V

of afluid

—

^{in this case, thefluidthatpasses} through the cross sectioneach

second—

arenotindependentquantitiesbut are relatedthroughthefluid density, p:

m /V = m/V

(3.2-1)

Thus, the density of^afluid can be used to convert a

known

volumetricflow ^rate

of

^aprocess streamtothe

mass

_flowrate of^thatstream orvice versa.

The mass

flowratesofprocessstreams

must

be

known

for

many

processcalculations, but

it is frequently

more

convenient to

measure

volumetric flow rates.

A common

procedure is

therefore to

measure V and

calculate

m from V and

the density of thestreamfluid.

TEST

YOURSELF

(Answers,

_p.

655)

0.659

g/cm

³₎ ina pipe is 6.59g/s.

What

is the volu- 1.

The mass

flowrate of

n-hexane

_(p

metricflowrateof the

hexane?

2.

The

volumetric flowrate of

CC1

_{4 (p}

=

1.595 g/cm³) in apipe is 100.0

cm

³/min.

What

^is

the

mass

flowrateof the

CC1

₄?

3.

Suppose

a gasisflowingthroughacone-shapedpipe.

How do

the

mass

flowrates of the gas attheinlet

and

outlet

compare? (Remember

the lawof conservation of mass.) Ifthe density of the gasis constant,

how do

thevolumetric flowrates atthese

two

points

compare? What

ifthe density decreases

from

inlettooutlet?

3.2b Flow Rate Measurement

Equipment Encyclopedia process parameters flowratemeasurement

A

^{flowmeter is} a device

mounted

in a process line thatprovides acontinuous reading of the flowrate intheline.

Two commonly

used

flowmeters—

therotameter

and

theorifice

meter—

are

shown

schematicallyin Figure 3.2-1. Perry's

Chemical

Engineers'

Handbook,

_{pp. 5-7} through 5-17, describes

many

others.

(6)

FIGURE

^3.2-1 Flowmeters: (a)rotameterand(b)orificemeter.

Variableswhosesymbolsincludea dot(•) arerates;forexample,

m

ismassflow rateand

V

isvolumetricflow rate.

3.3

Chemical Composition

47

The

rotameter is a tapered ^vertical tube containing a ^float; the larger the flow rate, the higherthefloat rises in thetube.

The

orifice

meter

is

an

obstructionin the flowchannelwitha

narrow

opening through

which

the fluidpasses.

The

fluidpressuredrops (decreases)

from

the

upstream

sideof theorificetothe

downstream

side;the pressure

drop

(which

may

be

measured

with a

number

ofdevices, including a differential

manometer, which

^isdiscussed in the next section)varieswiththe flow

rate—

the greater the flow^rate,the larger the pressure drop.

Problems

at the

end

of this chapter ^illustrate the calibration

and

use of both types of flowmeters.

1.

A

^{steadily flowing}

^steam

^{ofwater is funneledinto a graduated} cylinder for exactly30s,

during

which

time 50

mL

^iscollected.

^What

^{isthevolumetricflowrate of thestream?}

^The

mass

flow rate?

2.

What

isarotameter?

An

^{orifice meter?}

3.

A

^rotametercalibration curve (flow^rate versus floatposition) obtained using a liquidis

mistakenly used^to

measure

agas flowrate.

Would you

expectthe gas flow ratedetermined in this

manner

tobe too high ortoo low?