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CHAPTER 3

Volumetric Properties of Pure Fluids: Part 1. CHAPTER 3. Norhaniza binti Yusof. Faculty of Chemical Engineering Universiti Teknologi Malaysia, 81310 UTM Johor, Johor Bahru , Malaysia. Chemical Reaction Engineering Group, Universiti Teknologi Malaysia. Topic Outcomes.

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CHAPTER 3

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  1. Volumetric Properties of Pure Fluids: Part 1 CHAPTER 3 NorhanizabintiYusof Faculty of Chemical Engineering Universiti Teknologi Malaysia, 81310 UTM Johor, Johor Bahru, Malaysia Chemical Reaction Engineering Group, Universiti Teknologi Malaysia

  2. Topic Outcomes Chemical Reaction Engineering Group, Universiti Teknologi Malaysia

  3. Scope of Lecture • Introduction to volumetric properties • PVT behaviour of pure substances • Volumetric properties from equations of state Chemical Reaction Engineering Group, Universiti Teknologi Malaysia

  4. PVT Behavior of Pure Substances Chemical Reaction Engineering Group, Universiti Teknologi Malaysia

  5. PT Diagram for Pure Substances Fliq = 2 (divariant) 3 Fluid region A DOF for the system No. of species Pc Liquid region C Fusion curve Vaporization curve Fvap. curve = 1 (univariant) Pressure • F= 2 –  + N No. of phases Independent variables (T, P) B Solid region Gas region Ftp = 0 (invariant) Triple point Supercritical T > Tc Vapor region 2 Sublimation curve 1 Temperature Tc Note: DOF, degree of freedom Chemical Reaction Engineering Group, Universiti Teknologi Malaysia

  6. PV Diagrams for Pure Substances In a PV diagram, the phase boundaries becomes areas/region Liq., liq./vap. regions & vap. with isotherms Sol., liq. & gas regions Single-phase (sat.) vap. at cond. T Triple point Single-phase (sat.) liqs. at boiling T Subcritical T → consist of 3 segments Note: Sol., solid; liq., liquid; vap., vapor; sat., saturated; cond., condensation; T, temperature

  7. PVT Surface for a Real Substance P P CP CP Liquid Solid Gas T Vapor T Triple line V Note: CP., critical point Chemical Reaction Engineering Group, Universiti Teknologi Malaysia

  8. PVT Surface (Cont.) Constant temperature line Chemical Reaction Engineering Group, Universiti Teknologi Malaysia

  9. Pure Substances Phase • Any fluid could be in one of below conditions : • Single-phase liquid • Single-phase gas • 2-phases, liquid& gas Generally, if the system is at any T or P; *For H2O, Tc= 374 oCand Pc = 220 bar Note: Tc, critical temperature, Pc, critical pressure, Tbpt, biling point temperature

  10. Determination of Pure Substances Phase • Boiling point and vapor pressure could be obtained from : • Antoine equation • Cox Chart E.g.: Antoine equation for H2O; log10 P (mmHg) = 7.96681 – 1668.21 / (T oC + 228) • Liquid phase: V is almost stable and not depend on P and T. • Gas phase:V of pure substance is depend on T and P. • Mathematic relations between P, V, T of pure substance is called as EQUATION OF STATE Chemical Reaction Engineering Group, Universiti Teknologi Malaysia

  11. Single-Phase Region • From the regions of the diagram (PV) where a single phase exists, • implies a relation connecting P, V & T →express by function equation f (P, V, T ) = 0 PVT equation of state (EoS) Simplest EoS → ideal gas law • Valid for low P • Will be discussed later PV = RT • To solve the equation, • V = f (P , T ) or P = f (V , T) or T= f (P , V) Note: P, pressure; EoS; equation of state Chemical Reaction Engineering Group, Universiti Teknologi Malaysia

  12. Example V = V (P, T) V = f (P, T) Hence, Partial derivative in the equation Integration Divide by V Volume expansivity,  • For liquid phase • incompressible fluid  and  are very small ( ≈ 0). Isothermal compressibility,  Chemical Reaction Engineering Group, Universiti Teknologi Malaysia

  13. Notes • Gas • Ideal gas • Non-ideal gas • Equations of state (from PVT data). • Generalized correlations • Equations of state • Theorem of corresponding states PV = ZRT Valid at low pressure Z = 1 Chemical Reaction Engineering Group, Universiti Teknologi Malaysia

  14. Equation of State • Equation of State • Ideal Gas Equation • Virial Equation • Cubic Equation of State Van Der Waals Redlich/ Kwong Redlich/ Kwong/ Soave Peng/ Robinson Generic Vapor & Vapor-Like Roots Liquid & Liquid-Like Roots Chemical Reaction Engineering Group, Universiti Teknologi Malaysia

  15. Virial Equations of State Chemical Reaction Engineering Group, Universiti Teknologi Malaysia

  16. Virial Equations (Gas Phase) A useful auxiliary thermodynamic property is defined by the equation Compressibility factor There are 2 types of virial equations / virial expansions : B’, C’, D’, B, C, D etc. →virial coefficients • Only depend on T. • Obtained from PVT data. Note: T, temperature Chemical Reaction Engineering Group, Universiti Teknologi Malaysia

  17. Application of the Virial Equations CH4 Low to moderate P(P < 15 bar) 2 terms High Pup to ≈50 bar (below the Pc) Truncated to 3 terms Note: P, pressure; Pc, critical pressure Chemical Reaction Engineering Group, Universiti Teknologi Malaysia

  18. The Ideal Gas • Virial expansion arise on account of molecular interaction. • Term B/V arises on account of interactions between pairs of molecules • The C/V2 term, on account of 3-body interactions. • For ideal gas interaction molecular is assume not existed. • As the P of real gas is reduced at constant T • →V & the contribution of the terms B/V, C/V2, etc.,  . • As P → 0, V becomes , then Z approaches unity, = 1 PV = ZRT (low P only) Chemical Reaction Engineering Group, Universiti Teknologi Malaysia

  19. Internal energy • Internal energy of a REAL GAS is a function of P & T • P dependency is the result of forces between the molecules. • If such forces did not exist (IDEAL GAS behavior) → the internal energy of gas depends on T only U = U(T) (Ideal gas) Chemical Reaction Engineering Group, Universiti Teknologi Malaysia

  20. Property Relations for an Ideal Gas • Heat capacity at constant V(CV) is a func. of T only • Enthalpy (H) is a func. of T only • Heat capacity at constant P (CP) is a func. of T only • Relation between CP and CV This equation does not imply that CP and CV are themselves constant for an ideal gas, but only that they vary with T in such a way that their difference is equal to R. Chemical Reaction Engineering Group, Universiti Teknologi Malaysia

  21. Equations for Process Calculations • For a unit mass or a mole of IDEAL GAS in any mechanically reversible closed-system process: • These equations may be applied to the following processes Chemical Reaction Engineering Group, Universiti Teknologi Malaysia

  22. Adiabatic Process • Isothermal process: • Isobaric process: • Isochoric process: Chemical Reaction Engineering Group, Universiti Teknologi Malaysia

  23. Adiabatic Process (Const. Heat Capacities) • Equations • Other expressions • Work • The process should be mechanically reversible! • Alternative forms Chemical Reaction Engineering Group, Universiti Teknologi Malaysia

  24. Example 1 (Ideal gas) One mol of an ideal gas with Cp = (7/2)R and Cv = (5/2)R expands from P1 = 8 bar and T1 = 600 K to P2 = 1 bar by each of the following paths: Constant volume Constant temperature Adiabatically Assuming mechanical reversibility, calculate W, Q, U, and H for each process. Chemical Reaction Engineering Group, Universiti Teknologi Malaysia

  25. Example 2 (Virial Equation) • Reported values for the virial coefficients of isopropanol vapor at • 200 C are: • B = – 388 cm3 mol–1 C = – 26000 cm6 mol–2 • Calculate V and Z for isopropanol at 200 C and 10 bar by: • The ideal-gas equation • Truncated virial equation to 3 terms Chemical Reaction Engineering Group, Universiti Teknologi Malaysia

  26. Tutorial 1 For methyl chloride at 100 C the 2nd and 3rd virial coefficients are: B = – 242.5 m3 mol –1 C = 25, 200 cm6 mol –2 Calculate the work of mechanically reversible, isothermal compression of 1 mol of methyl chloride from 1 bar to 55 bar at 100 C. Base calculations on the following forms of the virial equations: Where Chemical Reaction Engineering Group, Universiti Teknologi Malaysia

  27. N02 1-31012.00 – 1.00 P.MFeb 21, 2013 (Thu) Volumetric Properties of Pure Fluids: Part 2 CHAPTER 3 MohdAsmadi Bin Mohammed Yussuf Faculty of Chemical Engineering Universiti Teknologi Malaysia, 81310 UTM Johor, Johor Bahru, Malaysia Chemical Reaction Engineering Group, Universiti Teknologi Malaysia

  28. Scope of Lecture • Cubic equation of state • Generic cubic • Example & Tutorial Chemical Reaction Engineering Group, Universiti Teknologi Malaysia

  29. Cubic Equation of State Chemical Reaction Engineering Group, Universiti Teknologi Malaysia

  30. Generic Cubic Equation of State The simplest equations capable of representing both liquidand vaporbehavior but not for dual-phase condition. More accurate for a wide range of T and P Where b , θ , κ , λ and η are parameters which is general depend on Tand (for mixture) composition. Note: T, temperature; P, pressure; cubic equation of state ,15 < p < 50 bar

  31. Equation of State Parameters When in polynomial form : • Where • Pcand Tc → pressure and temperature in critical point • a and b → +ve constants. Chemical Reaction Engineering Group, Universiti Teknologi Malaysia

  32. Cubic Equation of State • Cubic Equation of State Van Der Waals Redlich/ Kwong Redlich/ Kwong/ Soave Peng/ Robinson Generic Vapor & Vapor-Like Roots Liquid & Liquid-Like Roots Chemical Reaction Engineering Group, Universiti Teknologi Malaysia

  33. van der Waals θ = a η = b A Generic Cubic EoS (previous slide) κ = λ = 0 Reduces to van derWaals EoS Will be discussed later Note: EoS; equation of state Chemical Reaction Engineering Group, Universiti Teknologi Malaysia

  34. Redlich/Kwong Equation, Solve by cubic equation solver. Chemical Reaction Engineering Group, Universiti Teknologi Malaysia

  35. V for Gas & Liquid Phases (RK) V for liquid phase V for gas phase Multiple & rearrange Solve by iteration method, use b as an initial Vo. Note: RK, Redlich/Kwong Chemical Reaction Engineering Group, Universiti Teknologi Malaysia

  36. Redlich-Kwong-Soave & Peng-Robinson Peng-Robinson Redlich-Kwong-Soave Acentric factor(App. B) Chemical Reaction Engineering Group, Universiti Teknologi Malaysia

  37. Parameters for all Cubic Equation of State Note: Please text book p. 98 Chemical Reaction Engineering Group, Universiti Teknologi Malaysia

  38. Generic Cubic Vapor & Vapor-Like Roots of the Generic Cubic EoS • Solution for V may be by • Trial • Iteration • With the solve routine of a software package Initial estimate for V is the ideal-gas value → Chemical Reaction Engineering Group, Universiti Teknologi Malaysia

  39. Dimensionless Quantities An equation for Z equivalent to equation above is obtained through substitute 2 dimensionless quantities leads to simplification Chemical Reaction Engineering Group, Universiti Teknologi Malaysia

  40. Determination of Z Therefore, Iteration Start with Z = 1 → Substituted on the right side. Calculated Z → Returned to the right side Process continues to convergence → yields the volume root through V = ZRT/P Final Z Chemical Reaction Engineering Group, Universiti Teknologi Malaysia

  41. Generic Cubic (Cont.) Liquid & Liquid-Like Roots of the Generic Cubic EoS V in the numerator of the final fraction to give : Starting value V = b. An equation for Z equivalent to the equation above : Chemical Reaction Engineering Group, Universiti Teknologi Malaysia

  42. Iteration For iteration Z = β. Once Z is known, the volume root is • EoS which express Z as a function of Tr& Pr • Called generalized Their general applicability to all gases & liquids. Note: EoS; equation of state Chemical Reaction Engineering Group, Universiti Teknologi Malaysia

  43. A Generic Cubic Equation of State An important class of cubic equations results from the preceding equation with the assignments η = b θ = a ( T ) κ = ( є + σ ) b λ = є σ b2 • ε & σare pure numbers • →same for all substances • a(T) is specific to each EoS Note: EoS; equation of state Chemical Reaction Engineering Group, Universiti Teknologi Malaysia

  44. Determination of EoS Parameters Suitable estimates of EoS parameters are usually found from values for the critical constantsTc and Pc→ critical isotherm exhibits a horizontal inflection at the critical point! At critical point ( Pc, Tc, Vc& Zc) : ReplacePc, Tc and Vc From van der Waals equation : Then, Note: EoS; equation of state Chemical Reaction Engineering Group, Universiti Teknologi Malaysia

  45. EoS Parameters (Cont.) An analogous procedure may be applied to the generic cubic equation Previous slide Yielding expressions for parameters • Ω and Ψ • Pure numbers, • Independent of substance • Determined for a particular EoS from the values assigned to ε & σ Chemical Reaction Engineering Group, Universiti Teknologi Malaysia

  46. EoS Parameters (Cont.) This result may be extended to T other than the critical by Introduction of a dimensionless function • unity at the Tc • Function α(Tr) : an empirical expression • specific to a particular equation of state. Note: Tc; critical temperature Chemical Reaction Engineering Group, Universiti Teknologi Malaysia

  47. Theorem of Corresponding States All fluids, when compared at the same Tr& Pr, have approximately the same Z, and all deviate from ideal-gas behavior to about the same degree . This theorem is very nearly exact for the simple fluids (Ar, Kr, Xe). However, systematic deviations are observed for more complex fluids. Note: Z; compressibility factor ; Tr, reduce temperature; Pr, reduced pressure

  48. Acentric Factor (Cont.) Approximate T dependence of the reduced vapor pressure Chemical Reaction Engineering Group, Universiti Teknologi Malaysia

  49. Acentric Factor (Cont.) Therefore, an acentric factor is introduced as follow: Acentric factor At Tr = 0.7 →= 0 (Ar, Kr & Xe) This value of  can be determined for any fluid from Tc, Pc and a single vapor-pressure measurement made at Tr = 0.7 App. B lists the values of ω and the critical constants Tc, Pc, and Vc for a number of fluids. Chemical Reaction Engineering Group, Universiti Teknologi Malaysia

  50. Acentric Factor (Cont.) All fluids having the same value of ω, when compared at the same Tr& Pr, have about the same value of Z, and all deviate from ideal-gas behavior to about the same degree. w Chemical Reaction Engineering Group, Universiti Teknologi Malaysia

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