1. CHAPTER 4 single-phase systems
Sem 1, 2015/2016
ERT 214 Material and Energy Balance / Imbangan Bahan dan Tenaga

2. 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)
Handbook or database
- Perry’s Chemical Handbook, CRC Handbook of Chemistry & Physics, TRC Database in Chemistry & Engineering, etc
Estimation using empirical correlations
Experimental work

3. Liquid and Solid Densities
When you heat a liquid or a solid it normally expands (i.e., its density decreases).
In most process applications, however, it can be assumed with little error that solid and liquid densities are independent of temperature.
Similarly, changes in pressure do not cause significant changes in liquid or solid densities; these substances are therefore termed incompressible.
To determine the density of a mixture of liquids or a solution of a solid in a liquid is from experimental data.
Perry's Chemical Engineers' Handbook provides data for mixtures and solutions of a number of substances on pp through and lists additional sources of data on p. 2-99

4. In the absence of data, the density of a mixture of n liquids (AI, A2,…. An) can be estimated from the component mass fractions [xi] and pure-component densities [σ1] in two ways. First, we might assume volume additivity-that is, if 2 mL of liquid A and 3 mL of liquid B are mixed, the resulting volume would be exactly 5 mL. Making this assumption and recognizing that component masses are always additive leads to the formula Second, we might simply average the pure-component densities, weighting each one by the mass fraction of the component:

5. 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 at low pressure and high temperature
Derived from the kinetic theory of gases by assuming gas molecules:
have a negligible volume;
Exert no forces on one another;
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 liters, whether the gas is argon, nitrogen, mixture of propane and air, or any other single species or mixture of gases

6. Ideal Gas Equation of State
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, T
T = absolute temperature
Ideal gas equation of state can also be written as
Which ; specific molar volume of gas.
Unit for gas constant, R or

7. Ideal Gas Equation of State
The ideal gas equation of state is an approximation.
It works well under some conditions = at temperatures above about 0oC and pressures below about 1 atm-but at other conditions its use may lead to substantial errors.
Here is a useful rule of thumb for when it is reasonable to assume ideal gas behavior.
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 P (absolute), T (absolute), n or V
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%

8. Example: The Ideal Gas Equation of State
One hundred grams of nitrogen is stored in a container at 23.0oC and 3.00 psig.
Assuming ideal gas behavior, calculate the container volume in liters.
The ideal gas equation of state relates absolute temperature, absolute pressure, and the quantity of a gas in moles. We therefore first calculate
and (assuming Patm = 14.7 psia) P = 17.7 psia. Then from the ideal gas equation of state
Unfortunately, the table of gas constants at the back of this book does not list the value of R with this particular set of units. In its absence, we use an available value and carry out the necessary additional unit conversions.
14x2=28
T=273+23=296K
Pabs=Patm+Pgauge =
=17.7 psig

9. 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

10. Standard Conditions for Gases
System Ts Ps Vs ns Vs (in STP)
SI 273K 1atm m3 1 mol m3/kmol
CGS 273K 1atm L 1 mol L/mol
American 492˚R 1atm ft lb-mole ft3 /lb-mole
Engineering

11. Example: Conversion from Standard Conditions
Butane (C4HIO ) at 360°C and 3.00 atm absolute flows into a reactor at a rate of 1100 kg/h. Calculate the volumetric flow rate of this stream using conversion from standard conditions.
Solution:
As always, molar quantities and absolute temperature and pressure must be used.
C4H10 = 12(4)+1(10)
=58 kg/kmol
T= = 633K

12. Example: Effect of T and P on Volumetric Flow Rates
Ten cubic feet of air at 70°F and 1.00 atm is heated to 610°F and compressed to 2.50 atm. What volume does the gas occupy in its final state?
Solution:
Let 1denote the initial state of the gas and 2 the final state. Note that n1 = n2 (the number of moles of the gas does not change). Assume ideal gas behavior.
P1V1=nRT1
P2V2=nRT2

13. Example: Standard and True Volumetric Flow Rates
The flow rate of a methane stream at 285°F and 1.30 atm is measured with an orifice meter. The calibration chart for the meter indicates that the flow rate is 3.95 X 105SCFH.
Calculate the molar flow rate and the true volumetric flow rate of the stream.
SOLUTION
Recall that SCFH means ft3(STP)/h.
Standard cubic feet (SCF)
Note that to calculate the molar flow rate from a standard volumetric flow rate, you don't need to know the actual gas temperature and pressure.
The true volumetric flow rate of the methane is calculated using this method
(T1 = 492°R, P1 = 1.0 atm, V1 = 3.95 X lO5 ft3/h) to actual conditions (T2 = 745°R, P2 = 1.30 atm, V2= ?). We therefore obtain
T (K) = T (˚ C)
T (˚R) = T (˚ F)
T (˚ R) = 1.8T (K)
T (˚ F) = 1.8T (˚ C) + 32
The ideal gas equation of state is an approximation.
It works well under some conditions- generally speaking, at temperatures above about 0oC and pressures below about 1 atm-but at other conditions its use may lead to substantial errors.
285oF= =745oR
At ideal gas condition, T=0oC and P=1atm So
T=0oC=273.15K=1.8x273.15=492oR

14. 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

15. 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

16. Example: Material Balance on an Evaporator Compressor
Liquid acetone (C3H60) 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 a temperature of 325°C. The partial pressure of acetone in this stream is Pa = 501 mm Hg. Atmospheric pressure is 763 mm Hg.
What is the molar composition of the stream leaving the compressor?
What is the volumetric flow rate of the nitrogen entering the evaporator if the temperature and pressure of this stream are 27°C and 475 mm Hg gauge?

17. What is the molar composition of the stream leaving the compressor?
Liquid acetone (C3H60) 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 a temperature of 325°C. The partial pressure of acetone in this stream is Pa = 501 mm Hg. Atmospheric pressure is 763 mm Hg.
What is the molar composition of the stream leaving the compressor?
SOLUTION
Basis: Given Feed Rates
Assume ideal gas behavior. Let n1, n2, ... (mol/min) be the molar flow rates of each stream
Atmospheric pressure is 763 mm Hg.

18. What is the molar composition of the stream leaving the compressor?
Atmospheric pressure is 763 mm Hg.
You should be able to examine the flowchart and see exactly how the solution will proceed.
Calculate n2 (from the given volumetric flow rate and a tabulated density of liquid acetone), n3 (from the ideal gas equation of state), and Y4 (= Pa/P).
2. Calculate n4 (overall acetone balance), n1 (overall mass balance), and V1 (ideal gas equation of state).

19. SG = substance/ref Calculate Molar Flow Rate of Acetone
What is the molar composition of the stream leaving the compressor?
Atmospheric pressure is 763 mm Hg.
Calculate Molar Flow Rate of Acetone
Calculate n2 (from the given volumetric flow rate and a tabulated density of liquid acetone),
From Table B.1 in Appendix B, SG(acetone)= so
So, density of liquid acetone is g/cm3 (791 g/L), so that
SG = substance/ref
(4˚C) = g/cm3
MW acetone=58.08 g/mol

20. Appendix B (Page 628)

21. What is the molar composition of the stream leaving the compressor?
Atmospheric pressure is 763 mm Hg.
Determine Mole Fractions from Partial Pressure
Calculate Y4 from (= Pa/P).
In the stream leaving the compressor,

22. What is the molar composition of the stream leaving the compressor?
Calculate n3 from STP Information
Calculate n3 (from the ideal gas equation of state)using STP
System Ts Ps Vs ns Vs (in STP)
SI 273K 1atm m3 1 mol m3/kmol
CGS 273K 1atm L 1 mol L/mol
American 492˚R 1atm ft lb-mole ft3 /lb-mole
Engineering

23. What is the molar composition of the stream leaving the compressor?
Calculate n4 from overall acetone mole balance

24. What is the molar composition of the stream leaving the compressor?
Overall Mole Balance
Calculate n1 (from overall mole balance n1 + n2 + n3 = n4

25. 2. What is the volumetric flow rate of the nitrogen entering the evaporator if the temperature and pressure of this stream are 27°C and 475 mm Hg gauge?
Calculate V1 (from ideal gas equation of state)

26. 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

27. Virial Equation of State
Expresses the quantity PV/RT as a power series in the inverse of specific volume.
Virial equation of state
B,C,D- second, third, fourth virial coefficient respectively (function of TEMPERATURE)
Truncated virial equation at the second term of yields
Tr=T/Tc
ω – acentric factor from Table 5.3-1
Tc,Pc from Table B.1

28. Example: The Truncated Virial Equation
Two gram-moles of nitrogen is placed in a three-liter tank at °C.
Estimate the tank pressure using the ideal gas equation of state and then using the virial equation of state truncated after the second term.
Taking the second estimate to be correct, calculate the percentage error that results from the use of the ideal gas equation at the system conditions.
SOLUTION:
IDEAL GAS EQUATION STATE

29. Two gram-moles of nitrogen is placed in a three-liter tank at -150.8°C.
Estimate the tank pressure using the ideal gas equation of state and then using the virial equation of state truncated after the second term.
Taking the second estimate to be correct, calculate the percentage error that results from the use of the ideal gas equation at the system conditions.
SOLUTION:
VIRIAL EQUATION OF STATE
The virial equation solution procedure is as follows:
1. Table B.1 ====> (Tc)N2 = K, (PC)N2 = 33.5 atm
2. Table ====> ωN2 = 0.040
3. Tr = T/Tc = 122.4K/126.2 K = 0.970

30. Appendix B (Table B.1)

31. Taking the second estimate to be correct, calculate the percentage error that results from the use of the ideal gas equation at the system conditions.
Pideal= 6.73 atm
P=6.19 atm

32. 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 state
Van der Waals equation of state
Soave-Redlich-Kwong (SRK) equation of state

33. Van der Waals Equation of State
In the van der Waals derivation, the term a/V2 accounts for attractive forces between molecules.
b is a correction accounting for the volume occupied by the molecules themselves

34. Soave-Redlich-Kwong (SRK) equation of state
where the parameters a, b, and α are empirical functions of the critical temperature and pressure (Tc and Pc from Table B.1),
the Pitzer acentric factor (ω from Table 53-1) and the system
temperature.
The following correlations are used to estimate these 3 parameters

35. Example: The SRK Equation of State
A gas cylinder with a volume of 2.50 m3 contains 1.00 kmol of carbon dioxide at T = 300 K. Use the SRK equation of state to estimate the gas pressure in atm.
SOLUTION:
The specific molar volume is calculated as

36. From Table B. 1, Tc = 304. 2 K and Pc = 72. 9 atm, and from Table 5
From Table B.1, Tc = K and Pc = 72.9 atm, and from Table 5.3-1, w =
The parameters in the SRK equation of state are evaluated using Equations through :

38. Compressibility Factor Equation of State
If z=1, equation become ideal gas equation of state
Value of z is given in Perry’s Chemical Engineering Handbook pg
Alternatively; can use generalized compressibility chart
Figure – generalized compressibility chart
Fig to Fig – expansion on various region in Fig

39. Step to Read Compressibility Factor
Find Tc and Pc
If gas is either Hydrogen or Helium, determine adjusted critical temperature and pressure from Newton’s correction equation
Calculate reduce pressure and reduce temperature of the two known variables
Read off the compressibility factor from the chart

41. 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

42. Example: Kay’s Rule
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/moI using Kay's rule.

43. Pseudocritical temperature (Tc’)
Tc’= yATcA + yBTcB +……
Pseudocritical pressure (Pc’)
Pc’= yAPcA + yBPcB +……
A mixture of 75% H2 and 25% N2 (molar basis) is contained in a tank at 800 atm and
-70°C.

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