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CENG 131 CENG 131 Lecture 3. Equation of States (3 h) Lecture 3. Equation of States (3 h) Learning Objectives: Learning Objectives: (1) PVT data (1) PVT data (2) Virial equations (2) Virial equations (3) Generalized compressibility-factor correlation (3) Generalized compressibility-factor correlation (4) Generalized virial-coefficient correlation (4) Generalized virial-coefficient correlation (5) Cubic equations of state (5) Cubic equations of state Learning Guides: Learning Guides: (1) Lecture handouts (1) Lecture handouts (2) Chapters 3.1-3.6 of Introduction to Chemical Engineering (2) Chapters 3.1-3.6 of Introduction to Chemical Engineering Thermodynamics 5th ed. (Smith, Van Ness, Abbott) Thermodynamics 5th ed. (Smith, Van Ness, Abbott)

Temperature-Entropy (TS) Diagram of Nitrogen

Experimental data are the most accurate source of information on the P-V-T states of a substance. Note: The data includes both liquid and gas phases.

Problem 10: 1 moke of nitrogen gas was compressed adiabatically from ambient pressure (1 bar) to 50 bar using a two stage compressor. What is the final volume and temperature of oxygen at the end of the compression. If the gas is cooled at constant pressure back to room temperature and further compressed isothermally to 100 bar, what is the final volume of the nitrogen. (a) Assume that nitrogen is an ideal gas and use the ideal gas equations for the calculation. (b) Use the experimental data for oxygen to determine the actual state of the gas.

Virial Equations

Taylor expansion for f(p,T)

where the constant are evaluated experimentally:

at low pressure:

Acoustic Compressor

Virial equation is derived from experimental PVT data and is the next best thing to a TS-diagram. Use virial equations if P > 50 bar. Note: The data is only for gas phase unless specified.

Virial Equations

Z = PV = 1 + B'P + C'P2 + D'P3 + .... RT Z = PV = 1 + B + C + D + .... RT V V2 V3

Problem 11: The high heat capacity and latent heat associated with water and its availability made it an ideal working fluid. Steam driven piston and cylinder assembly is the corner stone of industrial revolution. Therefore significant amount of work were done to determine the exact behavior of steam. (a) Determine the final volume of 1 mole of steam compressed isothermally from 1 bar to 100 bar. (b) What is the work needed and the heat that has to removed? Steam: B = -53.4 cm3mol-1, C = 2,620 cm6mol-2, D = 5,000 cm9mol-3

B' = B RT

Compounds Methane Ethane Ethylene Methyl chloride Sulfur hexaflouride Steam

2 C' = C-B 2 (RT)

3 D' = D-3BC +3 2B (RT)

B(cm3mol-1)

-53.4 -156.7 -140 -242.5 -194 -152.5

C(cm6mol-2)

2620 9650 7200 25,200 15,300 -5800

Virial equation is derived from experimental PVT data and is the next best thing to a TS-diagram. Use virial equations if P > 50 bar. Note: The data is only for gas phase unless specified.

Friday Discussion Problem

(1) Molecular basis of virial equation of state (2) Experimental methodology and setup for determining PVT data of a pure substance (3) List 5 characteristics that you would like to have in a working fluid and why?

Generalized Compressibility Factor Correlation

Z = PV = Z0 + Z1 RT

Z0 and Z1 are read from a table as a function of Tr and Pr Tr = T/Tc Pr = P/Pc

Generalized Virial Coefficient Correlation

BP P Z = 1 + BP = 1 + RTc Tr RT c r BPc = B0 + B1 RTc B0 = 0.083 - 0.422 Tr 1.6 B1 = 0.139 - 0.172 Tr 4.2

Tr = T/Tc Pr = P/Pc

Problem 11: The high heat capacity and latent heat associated with water and its availability made it an ideal working fluid. Steam driven piston and cylinder assembly is the corner stone of industrial revolution. Therefore significant amount of work were done to determine the exact behavior of steam. Determine the final volume of 1 mole of steam compressed isothermally (T= 773K) from 1 bar to 100 bar using (a) generalized compressibility correlation (b) generalized virial-coefficient correlation

Cubic Equation of States

Redlich-Kwong Equations

Equation of States

PVT Equation PVT table PVT chart Virial Equation 4-terms 3-terms 2-terms 1-term (i.e., ideal gas equation) All < 100 bar < 50 bar < 10 bar All < 100 bar < 50 bar < 10 bar High High-Moderate Moderate Low Moderate Moderate Moderate Low Very difficult Difficult Moderate Easy Moderate ModerateDifficult ModerateDifficult Easy Pressure Range All All Accuracy Very high High Computation Easy Easy

P = RT V-b a= b=

a 0.5V(V+b) T

0.42748 R2Tc2.5

Pc

0.08664 RTc

Pc

Tr = T/Tc Pr = P/Pc

Z-correlation Equation Virial coefficient correlation equation Redlich-Kwong Equation Ideal Gas Equation

Vapor volume

Liquid volume

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