PTT108 MATERIAL AND ENERGY BALANCEChapter 5

Single Phase Systems

Madam Noorulnajwa Diyana

Introduction

Before carrying out a complete material balance, we usually need to determine various physical properties of materials in order to derive additional relationship among the system variables.

As an example we need the density to relate the volumetric flow rate to mass flow rate or vice versa.

3 ways to obtain the values of physical properties (such as density, vapor pressure, solubility, heat capacity, etc)1. Handbook or database

- Perry’s Chemical Handbook, CRC Handbook of Chemistry & Physics, TRC Database in Chemistry & Engineering, etc

2. Estimation using empirical correlations3. Experimental work

Density of Liquid and Solid

Temperature dependence: modest but sometimes important (liquid and solid expanded during heating and density decrease)

Pressure dependence: usually negligible (solid and liquid are incompressible with pressure).

2 methods to estimate the density of mixture which consist n liquid (n is number of different type of liquid) Method 1: Volume Additivity

Works best for mixture of liquid species with similar molecular structure

Method 2: Average Pure Component Densities

n

i i

ix

1

1

n

iiix

1

Ideal Gases

Equation of state Relates the molar quantity and volume of a gas to temperature and

pressure. Ideal gas equation of state

Simplest and most widely used Used for gas a low pressure and high temperature Derived from the kinetic theory of gases by assuming gas molecules

1. have a negligible volume;2. Exert no forces on one another;3. Collide elastically with the wall of container

or

The use of this equation does not require to know the gas species:1 mol of an ideal gas at 0˚C and 1 atm occupies 22.415 liters, whether the gas is argon, nitrogen, mixture of propane and air, or any other single species or mixture of gases

nRTPV RTnVP

Ideal Gas Equation of State

P = absolute pressureV = volume of the gasn = number of moles of gasR = gas constant which the unit depend on unit of P, V, n, TT = absolute temperature

Ideal gas equation of state can also be written as Which ; specific molar volume of gas.

Unit for gas constant, R

or

nRTPV

RTVP ˆnVV /ˆ

etemperaturmole

volumepressureRfor Unit

etemperaturmole

energyRfor Unit

Ideal Gas Equation of State Density of ideal gas (M is average molecular weight-refer back previous

chapter Eq. 3.3-7)

Rule of thumb for when it is reasonable to assume ideal gas behavior. Let Xideal be a quantity calculated using ideal gas equation of state (X can

be pressure, volume, temperature or mole). Error is estimated value is ε

Let’s say quantity to be calculate is ideal specific molar volume,

If error calculated satisfies this criterion, the ideal gas equation of state should yield an error less than 1%

RT

MP

V

M

ˆ

%100

true

trueideal

X

XX

V̂

gasesother mole)-lb/ ft (320 L/mol 20

gasesdiatomicmole)lb/ ft (80L/mol 5ˆ%13

3

idealVif

Class Discussion

Example 5.2-1

Standard Temperature and Pressure (STP)

A way to avoid the use of gas constant, R when using ideal gas equation For ideal gas at arbitrary temperature, T and pressure, P

For the same ideal gas at standard reference temperature, Ts and standard reference pressure, Ps (refer to STP).

Divide eq. 1 to eq. 2

Value of standard conditions (Ps, Ts, Vs) are known, above equation can be used to determine V for a given n or vice versa

Standard cubic meters (SCM) : m3 (STP) Standard cubic feet (SCF) : ft3 (STP) Let say 18 SCMH mean 18 m3 (STP)/h

nRTPV

sss RTVP ˆ

sssT

Tn

VP

PV

ˆ

Standard Conditions for Gases

System Ts Ps Vs ns VsSI 273K 1atm 0.022415 m3 1 mol 22.4 m3/kmolcgs 273K 1atm 22.415 L 1 mol 22.4 L/molEnglish 492˚R 1atm 359.05ft3 1 lb-mole 359.05 ft3/lb-mole

“ Saya pasti akan berjaya kerana saya telah

kehabisan benda-benda yang tidak berjaya”

Thomas Edison

Class Discussion

Example 5.2-2

Class Discussion

Example 5.2-3

Class Discussion

Example 5.2-4

Ideal Gas Mixture

Suppose nA moles of species A, nB moles of species B, nc moles of species C and so on, contained in a volume, V at temperature, T and pressure, P Partial pressure, pA

The pressure that would be exerted by nA moles of species A alone in the same total volume, V at the same temperature, T of the mixture.

Pure component volume, vA

The volume would be occupied by nA moles of A alone at the same total pressure, P and temperature, T of the mixture.

Ideal gas mixture Each of the individual species component and the mixture as whole

behave in an ideal manner

Pyp AA

Vyv AA

Ideal Gas Mixture

Dalton’s Law The summation of partial pressure of the component of an

ideal gas mixture is equal to total pressure

Amagat’s Law

Volume fraction = vA/V; percentage by volume (%v/v)= (vA/V )x 100%

For an ideal gas mixture, the volume fraction is equal to the mole fraction of the substance:

70% v/v C2H6 = 70 mole% C2H6

PPyyyppp CBACBA ....)(.....

VVyyyvvv CBACBA ....)(.....

Class Discussion

Example 5.2-5

Equation of State for Nonideal Gases

Critical temperature (Tc)- the highest temperature at which a species can exist in two phases (liquid and vapor), and the corresponding pressure is critical pressure (Pc)

Other definition: highest temperature at which isothermal compression of the species vapor results in the formation of a separate liquid phase.

Critical state- a substance at their critical temperature and critical pressure.

Species below Pc: Species above Tc- gas Species below Tc- vapor

Species above Pc and above Tc- supercritical fluids

Virial Equation of State Virial equation of state

B,C,D- second, third, fourth virial coefficient respectively

Truncated virial equation

Tr=T/Tc ω – acentric factor from Table 5.3-1

Tc,Pc from Table B.1

....ˆˆˆ

1ˆ

32

V

D

V

C

V

B

RT

VP

V

B

RT

VPˆ

1ˆ

6.102.4110

422.0083.0;

172.0139.0);(

rrc

c

TB

TBBB

P

RTB

Class Discussion

Example 5.3-1

Kebebasan tanpa tanggungjawab membawa

kehancuran

Cubic Equations of State

Refer as cubic equation because when the equation is expanded, it becomes third order equation for the specific volume

To evaluate volume for a given temperature and pressure using cubic equation of state, we need to do trial and error procedure.

Two famous cubic equation of statea) Van der Waals equation of stateb) Soave-Redlich-Kwong (SRK) equation of state

Van der Waals Equation of State

(a/V2) - account for attractive force between moleculesb - correction accounting for the volume occupied by the

molecules themselves

2ˆˆ V

a

bV

RTP

c

c

c

c

P

RTb

P

TRa

864

27 22

Soave-Redlich-Kwong (SRK) equation of state

)ˆ(ˆˆ bVV

a

bV

RTP

2

2

2

1561.055171.148508.0

)]1(1[/

086.40)(

42747.0

m

TmTTT

P

RTb

P

RTa

rcr

c

c

c

c

Class Discussion

Example 5.3-2

Compressibility Factor Equation of State

or

If z=0, equation become ideal gas equation of state

Value of z is given in Perry’s Chemical Engineering Handbook pg. 2.140- 2.150.

Alternatively; can use generalized compressibility chart Figure 5.4-1 – generalized compressibility chart Fig. 5.4-2 to Fig. 5.4-4 – expansion on various region in Fig. 5.4-

1

zRTVP ˆRT

VPz

ˆ

Step to Read Compressibility Factor

1. Find Tc and Pc2. If gas is either Hydrogen or Helium, determine adjusted critical

temperature and pressure from Newton’s correction equation

3. Calculate reduce pressure and reduce temperature of the two known variables

4. Read off the compressibility factor from the chart

c

cidealr RT

VPV

Pc

P

Tc

TTr

ˆ;Pr;

atmPPKTT cacc

ac 88

Class Discussion

Example 5.4-2

Nonideal Gas Mixtures

Kay’s Rule: estimation of pseudocritical properties of mixture as simple average of pure a component critical constants

Pseudocritical temperature (Tc’)

Tc’= yATcA + yBTcB +……

Pseudocritical pressure (Pc’)

Pc’= yAPcA + yBPcB +……

Pseudocritical reduced temperature (Tr’)

Tr’= T/Tc’

Pseudocritical reduce pressure (Pr’)

Pr’= P/Pc’

Compressibility factor for gas mixture, Zm P

RTzV mˆ

Class Discussion

Example 5.4-3

ANY QUESTION?