Chapter 5 Single Phase Systems Chemial engineering alulations rely heavily on the availability of physial properties of materials. There are three ommon methods used to find these properties. These inlude measuring them, looking them up in the available literature, and finally they an be estimated using physial models, that usually ome in the form of equations. 1. Properties of Pure Ideal Components Definitions Correlation: This is a mathematial fit of experimental data (usually using statistial methods). These are used either in plae of or in onjuntion with physially based models. Equations of State (EOS): These are orrelation or physially based equations that relate Pressure, Volume, and Temperature that exist inside a material (typially gas). These three thermodynami variables are easily measured and make up what is known as a PVT relationship. Inompressible: This term is used to speify the assumption of onstant density of a material. Inompressibility is losely followed by liquids and solids under many situations of interest. Density of Liquids and Solids Solutions and Mixtures. The density of a liquid or solid solution an be approximated by the following: ρ = n x i i= 1 ρ i (1) 1 ρ = n x i i= 1 ρi (2) 1
Here, x i is the mass fration of omponent i. The density of the solution, ρ, is found by adding ontributions from eah pure omponent density ρ i aording to its mass fration. The more you have of i, the more it affets the total average density. Equation 2 assumes that the total volume is the sum of the individual volumes; although this additive volume onept may make sense intuitively it is not neessarily true. Volume additivity is relatively aurate for solutions of liquids with similar moleular strutures. In general, the auray of equation 1 over equation 2 depends on the system onsidered. Example. What is the average density of a hexane/otane mixture omprised of 10 kg of hexane and 30 kg of otane? The density of hexane is 0.66 and that of otane is 0.703. Density of Gases/Ideal Gas Equation Gases are highly ompressible, and therefore the inompressibility assumption does not apply. EOS s then must inlude the pressure, temperature, and volume to aount for the variation of the density of a gas. The simplest of these models is the ideal gas equation of state: PV = nrt (3) where P V ( V &, Vˆ) n (n) & R T m M ρ Absolute Pressure Volume (Volumetri Flow Rate, Speifi molar volume [= volume of 1 mole of gas partiles]) Number of moles (molar flow rate) Ideal gas onstant Absolute temperature Mass Molar mass (moleular weight) Density 2
Equation 3 an be rewritten in other forms. These alternate forms may be useful depending on the situation; they are given below: P V & = nrt & (4) using V = V / t ˆ (5) using V ˆ = V n P V = RT P ρrt M = (6) using n = m M & ; n& = n / t The ideal gas EOS is based on the primary assumptions that There are no intermoleular fores between the gas partiles. The volumes of moleules are negligible (i.e. the gas partiles an be treated as point partiles). Understanding these assumptions helps to understand the utility and limitations of this EOS. For example the ideal gas assumption annot be used at higher pressures beause moleules will interat with eah other. Also, at lower temperatures there an be substantial error assoiated with the ideal gas equation, beause moleules do not have enough thermal energy to overome the interations between one another. Generally, the ideal gas EOS works well at pressures of 1 atm or less and temperatures greater than 0 o C (both onditions should be satisfied). The text desribes further tests that an be applied to validate the appliability of ideal gas behavior. Standard Temperature and Pressure (STP). Most often (always for this ourse), STP onditions are defined as 0 o C and 1 atm. Standard ubi meters (SCM) and standard ubi feet (SCF) refer to volumes of gas evaluated at STP onditions. The volume of one mole of ideal gas at STP is 22.415 L = 0.022415 m 3. The volume of 1 lb-mole of ideal gas at STP is 359.05 ft 3. Example 5.2-4. The flow rate of a methane stream at 285 o F and 1.30 atm is stated to be 3.95 10 5 SCFM. What are the molar flowrate and true volumetri flowrate of the stream? 3
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Ideal Gas Mixtures Imagine we have a gas mixture inside a ontainer. We an then define: Partial Pressure: This is the pressure, P i, that the moles of gas i in the gas mixture would exert if the moles of gas i was all that was present in the ontainer (at the same temperature T). Pure Component Volume: This is the volume, v i, that the moles of pure gas i would oupy at the total pressure P and temperature T of the mixture. In an ideal gas mixture, the various gas moleules do not interat. This means that the pressure exerted by eah omponent i is independent of the others; thus its partial pressure is given by P i V = n i RT Dividing the above by the ideal gas EOS as applied to the total mixture (whih is assumed to behave ideally), we get P i V / (PV)= n i RT / (nrt) or P i = y i P (4) Therefore, the partial pressure of gas i in an ideal gas mixture an be alulated from the total pressure P and its mole fration y i. Dalton s Law states that the sum of the partial pressures over all the gases present in a mixture equals the total pressure (this follows from equation (4) and the fat that y = 1). For example, for a mixture of three gases A, B and C: i p i + pb + pc = ( yap + ybp + yc P) P (5) A = There is an analogous law - Amagat s Law for the pure omponent volumes v i of an ideal gas mixture. Amagat s law states that the sum of the pure omponent volumes is equal to the total volume: va + vb + vc = V (6) 5
where v i is given by v i = y i V (7) Note that the volume fration of i, that is the ratio between the pure omponent volume of i and the total volume, is also equal to the mole fration, v i /V= y i. Thus, in an ideal gas mixture, the mole fration and volume fration of i are numerially equivalent. Example. Prove equation (6). 6
Non-ideal Gases Definitions Critial Temperature: This is the temperature, T, above whih a pure omponent an no longer oexist as a mixture of liquid and vapor phases. Critial Point (Critial State) of a Fluid: The ritial point for a fluid refers to the state when the fluid is at T and the orresponding ritial pressure, P. Suh a fluid an still oexist, at T and P, as a liquid and a vapor mixture. However, if the temperature inreases past T then oexistene is not possible and the fluid is referred to as a superritial fluid (for P > P ) or as a gas (for P < P ). Critial Pressure: This is the pressure, P, at the ritial point. Redued Pressure (P r ): This is the atual pressure of a fluid divided by its ritial pressure, P = P P. r Redued Temperature (T r ): This is the atual temperature of a fluid divided by its ritial temperature, T = T T. r Law of Corresponding States: This is an empirially based priniple that states that all fluids deviate from ideality in a similar fashion, when ompared at the same redued temperature T/T and redued pressure P/P. Pitzer Aentri Fator: This fator (symbol ω), is a parameter used in non-ideal equations of state that takes into aount the geometry and polarity of a moleule. Nonideal gases: Beause in reality moleules of a gas do interat, all real gases are nonideal. Even though the ideal gas EOS may be a very good approximation at low pressures and high temperatures, at higher pressures and/or lower temperatures the impat of intermoleular interations on gas behavior inreases. Then, methods for nonideal gases must be used to aount for the effet of these interations on the relationship between P, V, and T (and related properties suh as density) of a gas. Below, the following five methods will be introdued: 7
A. Virial Equation of State B. Van der Waals Equation of State C. Soave-Redlih-Kwong (SRK) Equation of State D. Compressibility Fator Equation of State E. Kay s Rule (for nonideal gas mixtures) Virial Equation of State The virial equation of state is an infinite power series in the inverse of the speifi molar volume: PVˆ RT B C D = 1+ + + ˆ ˆ 2 ˆ 3 V V V +L (8) Equation 8 has a rigorous theoretial basis. With B and all higher virial oeffiients set to 0, it redues to the ideal gas EOS. Most often, the virial EOS is trunated after the seond term to simplify to: PVˆ RT B 1+ Vˆ P RT = or V ˆ 2 V ˆ = B (9) B is alled the seond virial oeffiient. If P is the unknown, equation 8 or 9 is easily solved. If the speifi molar volume is the unknown, the trunated equation 9 an be solved using the quadrati formula. Note, however, that equation 9 loses auray for polar moleules. B an be estimated from the following orresponding states orrelations: B B B 0 1 0.422 0.083 = (10) 1.6 T r 0.172 0.139 = (11) RT P 4.2 T r = ( B 0 + ωb 1 ) (12) Our text lists values for the Pitzer aentri fator in table 5.3-1 and for the ritial pressure and temperature in Appendix B (table B.1). 8
Van der Waals Equation of State Starting with the ideal EOS, P = RT Vˆ two modifiations an be made to aount for non-ideal behavior: P RT a ˆ 2 V b Vˆ = (13) The b in the denominator aounts for the fat that real gas moleules do possess a volume; thus reduing the total volume available to the gas (what would happen if a Van der Waals gas is progressively ompressed?). The seond term on the right, involving a, aounts for moleular interations. A positive a implies that the gas moleules attrat one another. The effet of this attration is to redue the tendeny (and thus the pressure) of the gas to expand. The parameters a and b hange from one gas to another but are independent of temperature. They an be expressed as funtions of the ritial temperature and pressure: 2 27R T a = 64P 2 b RT 8P = (14) The Van der Waals EOS allows solution for P if T and Vˆ, and the ritial state, are known. If the equation were instead to be solved for Vˆ, a ubi equation would result whih an be solved by trial and error. Alternately, a rather lengthy analytial formula exists for the roots of a ubi equation that ould also be used. Thus, the Van der Waals EOS is an example of a ubi equation of state. Soave-Redlih-Kwong (SRK) Equation of State A more aurate ubi EOS is the empirial SRK equation: 9
RT P = V b Vˆ αa ( Vˆ + b) ˆ (15) The parameters of the SRK EOS are alulated from the following: a 2 R T 0.42747 P RT P 2 = (16) b 0. 08664 = (17) [ + m( 1 T )] 2 α = (18) 1 r 2 m = 0.48508 + 1.55171ω 0.1561ω (19) Example 5.3-3. A stream of propane flows at 100.0 kmol/h, at temperature 423 o K and pressure P. Using the SRK EOS, estimate the volumetri flowrate of the stream for P = 0.7 atm and P = 70 atm. Also, alulate the perentage differene in the volumetri flowrate between the values obtained from the SRK EOS and the ideal gas EOS. 10
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Compressibility Fator Equation of State The ompressibility fator Z for a gas is defined as Z = PV RT. Z serves as a measure of deviation from ideal behavior. If the gas is ideal, then Z = 1; otherwise, it has a value other than unity. If Z is known, it an be used to alulate P from T and Vˆ, or Vˆ from P and T, et., just as we did for an ideal gas using: ˆ or PV & =Z n& RT (20) P V = ZRT Where an Z values be found? One an appeal to the law of orresponding states. In partiular, we will assume that the ompressibility fator Z, for an arbitrary gas, depends predominantly on the proximity to its ritial state as aptured in the redued temperature T r and redued pressure P r. Then, a generalized ompressibility hart an be onstruted whih gives values of Z as a funtion of T r and P r (e.g. see Fig. 5.4-1 in the text). If we take a look at suh a hart, we ll notie that as a fluid approahes its ritial state it markedly deviates from ideal gas behavior. Strong deviations are also observed at low temperatures and high pressures. The proedure for using the generalized ompressibility hart is as follows: 1. Find the ritial pressure and the ritial temperature for the fluid of interest (note: if the fluid is hydrogen or helium these ritial values must be orreted see text). 2. Calulate the redued pressure and temperature (note: all temperatures used must be absolute!). 3. Use the ompressibility hart to determine Z. In some problems, the pressure or the temperature may be unknown, and instead the molar volume Vˆ is given. In that ase, you an instead alulate an ideal redued volume as follows: ˆ P Vˆ Vˆ ideal r = (21) RT 12
and use a generalized ompressibility hart that is expressed in terms of ideal Vˆ r instead of the unknown pressure or temperature (see Fig. 5.4-4 in the text). Kay s Rule Nonideal Gas Mixtures Above, we onsidered several methods for aounting for the nonideal behavior of real, but pure, gases. What about mixtures of nonideal gases? In general, this is a diffiult problem. Kay s Rule is a relatively simple method that works best for mixtures of nonpolar gases whose ritial temperatures and pressures are within about a fator of 2 of one another. Kay s Rule is an empirial mixing rule that uses data from generalized ompressibility harts. The proedure for using Kay s Rule is as follows: 1. Calulate the pseudoritial pressure P * and the pseudoritial temperature T * for the mixture, using (with orretions for H 2 and He if needed): P * = y APA + ybpb + yc PC +L (22) T * y T + y T + y T +L (23) = A A B B C C 2. Calulate the pseudoredued pressure and the pseudoredued temperature, using: * * P r = P P (24) * * T r = T T (25) If Vˆ for the mixture is known instead of T or P, you an alulate a ideal * * pseudoredued ideal volume, V ˆ r = P V RT, instead of the pseudoredued temperature or pseudoredued pressure. 3. Use the generalized ompressibility harts to determine Z m, the ompressibility fator for the mixture, making use of the pseudoredued values just as you would of the redued values in the ase of a pure fluid. The ompressibility fator EOS for a nonideal mixture is defined as for a pure fluid: PVˆ = Z RT (26) m 13
Final remark: All equations of state disussed above have inherent approximations, and like the ideal gas equation have ranges of validity and errors assoiated with them. 14