Chapter 4 Single-Phase System

Chapter 4 Single Phase System

Material and Energy Balances

Single-Phase Systems

Liquid and Solid

Densities

Ideal gas relationships

Equation of States

(EOS) for Nonideal Gas

The Compressibility

Factor EOS

Topic OutcomesAt the end of Chapter 4, you should:• Describe the meaning of incompressible fluids, the

term "equation of state", ideal gas behavior, standard specific volume, partial pressure, compressibility factors, z and the critical T and P.

• Calculate PVT properties using ideal gas equation of states.

• Carry out PVT calculations for gas using truncated virial EOS, van der Waals EOS, SRK EOS and the compressibility factor EOS with either tabulated compressibility factors or a generalized compressibility chart for a single species and Kay's rule for non-ideal mixture of gases.

Introduction Before carried 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.

3 way 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

Liquid and solid expanded during heating and density decreases. However, it is assumed that the densities are independent of temp.

Change in pressure also do not caused changes in solid and liquid densities. These substance is termed as incompressible.

2 methods to estimate the density of mixture consist n number of liquid. Method 1: Volume Additivity

Method 2: Average Pure Component Densities

n

i i

ix

1

1

n

iiix

1

Ideal Gases

Equation of state Equation that relates the molar quantity and

volume of a gas to temperature and pressure. Ideal gas equation of state

Simplest and most widely used for many engineering calculations involving gases at low pressure.

Used for gas a low pressure and high temperature. For high pressure and/or low temperature, more

complex equation of state for P,V and T calculation.

Ideal Gas Equation of State

From the assumptions that gas molecules: 1)have negligible volume, 2) no force on one another and 3) collide elastically with the wall container. The equation:

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-cont.

P = absolute pressure

V = volume of the gas

n = number of moles of gas

R = gas constant which the unit depend on unit

of P, V, n, TT = absolute temperature

nRTPV


Ideal gas equation of state can also be written as

Which ; specific molar volume of gas. Unit for gas constant, R

R= 83.14 cm3 bar mol-1 K-1 R= 8.314 J mol-1K-1

RTVP ˆ

nVV /ˆ

etemperaturmole

volumepressureRfor Unit

etemperaturmole

energyRfor Unit


Density of ideal gas is calculated as:

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

RT

MP

V

M

ˆ

%100

true

trueideal

X

XX

V̂

To calculate is ideal specific molar volume,

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

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

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

3

idealVif

Try YourselfExample 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

--------- (1) For the same ideal gas at standard reference

temperature, Ts and standard reference pressure, Ps (refer to STP).

-----------(2) Divide eq. 1 to eq. 2 gives

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

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) ^Vs= 22.4 m3 (STP)/kmol =22.4 L(STP)/mol=359 ft3 (STP)/lb-mole

Ideal Gas Mixture

Suppose nA moles of species A, nB moles of species B, nc moles of species C and so on, is behave in ideal manner and 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. Pyp AA

Ideal Gas Mixture The partial pressure of a component in an

ideal gas mixture is the mole fraction of that component times the total pressure.

The summation of partial pressure of the component of an ideal gas mixture is equal to total pressure (Dalton’s Law).

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

Ideal Gas Mixture-cont.

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.

Amagat’s Law

Vyv AA

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

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

Liquid acetone (C3H6O6) is fed at a rate of 400 L/min into a heated chamber, where it evaporates into a nitrogen stream. The gas leaving the heater is diluted by another nitrogen stream flowing at a measured rate of 419 m3 (STP) min. The combined gases are then compressed to a total pressure P= 6.3 atm gauge at temperature of 325˚C. the partial pressure of acetone ion this stream is pa=501 mmHg. Atmospheric pressure is 763 mmHg.

a) What is the molar composition of the stream leaving the compressor?

b) What is the volumetric flow rate of the N2 entering the evaporator if the temperature and pressure of this stream are 27˚C and 475 mmHg gauge?

Try This

Equation of State for Nonideal Gases

The ideal gas equation might be inaccurate for low pressure and high temp. conditions. Hence, more accurate equation will be introduced.

Either the ideal gas equation fits the PVT data for a species depends on system temperature and pressure which is critical temperature and critical pressure.

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).

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

Gas vs Vapor

At pressure low enough for the species to not be a liquid:

Vapor: a gaseous species below its critical temperature.

Gas: a gaseous species above its critical temperature.

Species above Pc and above Tc- supercritical fluids.

Virial Equation of State Virial equation of state

B,C,D- second, third, fourth virial coefficients

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

Cubic Equations of State

Refer as cubic equation because when the equation is expanded, it become 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

Try This

A gas cylinder with a volume of 2.5 m3 contains 1.00 kmol of carbon dioxide at T= 300 K. Use the Soave-Redlich-Kwong equation of state to estimate the gas pressure in atm.

Compressibility Factor Equation of State

orIf z=1, equation become ideal gas equation of state

The extent to which z differs from 1 is a measure of the gas is behaving nonideally.

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 Use Read Compressibility Factor

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

adjusted critical temperature and pressure form Newton’s correction equation

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

4. Read of 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 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

A mixture of 75% H2 and 25% N2 (molar basis) is contained in a tank at 800 atm and -70°C. Estimate the specific volume of the mixture in L/mol using Kay’s rule.

The End

Chapter 4 Single-Phase System

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