化工熱力學(國際班)課件 Ch2 Volumetric Properties of Fluids 流體的容積性質

1、Chapter 2 Volumetric Properties of Fluids 流體的容積性質 U, H and S are often evaluated with PVTthe Volumetric Properties of FluidsThe PVT relations are important2.1 PVT Behavior of Pure Substances純物質PVT行為Matter appears in 3 states: gasliquidsolid. The P-T graph: The phase rule:vaporgasIn single phase regi

2、on (單相區(qū)）：f=2, divariant On phase conversion line(相轉變線）: f=1,univariantAt the triple point(三相點）：f=0, invariantreThe P-V diagram At the critical point :re2.2 The State Equation of Fluids 流體狀態(tài)方程2.2.1 The Equation of State (EOS) for ideal gases:Pabsolute pressure, Vmmolar volume, Tin K 2.2.2 Virial Equa

3、tions 維里方程(Onnes, 1901):Or:If P0Ideal gas, P0.1MPaWhen P1.5MpaEx.2.1Isopropanol, T=200=473.15K, P=1MPa, B=-0.388m3/kmol, c=-0.026 m6/kmol2 . Vm=?Solution :a. With EOS for ideal gas：b. with eqn (2.8)c. with eqn(2.9)Iterative calculation :We set:由此獲得1901年諾貝爾物理學獎2.3 Cubic equation of state 立方型狀態(tài)方程2.3.1

4、 van der Waals Eqn. 范德華方程 (J D van der Waals 1873 )orAt the critical point :Cubic eqn, 3 roots, 1 real and 2 complex (T=Tc)At the critical point, 3 equal real roots (Vc)When TTc, in L-G 2 phase region, 3 different real roots, VminVL, VmaxVG, the middle meaningless.Solve for a and b:and All the subst

5、ance have the same Zc-not the case!MaterialH2HeN2ArO2CO2H2OZc0.3320.3200.2920.2920.2920.2870.224van der Waals eqn, when applied near critical points, forms evident deviations.Ex.Find the pressure necessary to compress CO2 at 0 isothermally to a density of 80kg/m3 with van der Waals eqn ( the experim

6、ental value :3.09106Pa)。 Solution:The critical values for CO2 :=3.269106Pa 2.3.2 The RK Eqn. (Otto Redlich and J S Kwong, 1949)orAt the critical point :Vapor volumes: rearrange RK eqn.:Liquid volumes: rearrange RK eqn. in normal polynomial form:The initial value can be chosen as the result of ideal

7、gas law.b would be a suitable initial value.reEx. resolve the above Ex with R-K eqn.Better than van der Waals eqnFind the pressure necessary to compress CO2 at 0 isothermally to a density of 80kg/m3 with R-K eqn ( the experimental value :3.09106Pa)。 2.3.3 The Soave-Redlich-Kwong (SRK) Eqn.Ex. 2.3Cal

8、culate Z for N2 at 0 and 101.325MPa. The experimental measurement is 2.0685。 Solution ：find the critical values (App B p257):With R-K eqn:By iterative procedures:15Then:With SRK eqn:By iterative procedures:Then:Better than R-K eqnThe SRK eqn:Numerical solution of nonlinear equation with Newton metho

9、dA nonlinear equation :Any function can be deployed into Taylor series:Terminate the series after the first derivative:Solve the above eqn for :For the(n+1)s iteration:Numerical derivative:The iteration continues until:2.3.4 The Peng-Robinson (P-R) equation2.3.5 A generic cubic equation of stateAll

10、of cubic equation of state are the special cases of the equation:Table 2.1 Parameters Assignments for Equations of StateEq. of State (Tr)ZcvdW (1873)10027/641/83/8R-K (1949)Tr-0.5100.427480.086641/3SRK (1972)SRK (Tr; )100.427480.086641/3P-R (1976)P-R (Tr)1+20.51-20.50.457240.077790.30740SRK (Tr; ) =

11、 1 + ( 0.480 + 1.574 - 0.176 2 ) ( 1- Tr1/2 ) 2P-R (Tr; ) = 1 + ( 0.37464 + 1.54226 - 0.26992 2 ) ( 1- Tr1/2 ) 22.4 Generalized Correlations of Gases氣體PVT關系的普遍化關聯Substitute into van der Waals eqn:All the gases behave similarly at the critical points Reduced temperature 對比溫度Reduced pressure 對比壓力Reduc

12、ed molar volume 對比摩爾體積Van der Waals theorems of corresponding state2.4.1 Theorem of corresponding statesThe RK Eqn.Multiply by V/RT :Used in iterative calculation2.4.2 Generalized cubic EOSGeneralized form of SRK eqnreEx. 2.5i-C4 vapor : T=360K, P=1.541MPa, Z=? (with generalized R-K, SRK eqn)Solutio

13、n :for i-C4, Tc=408.1, Pc=3.65MPa, =0.176With R-K eqn:With 8 iterative calculations, Z=0.7449，the deviation is 3.85。 With SRK eqn:SRK2.4.3 Z graph with Tr and Pr as parameters 兩參數普遍化壓縮因子圖With 9 iterative calc: Z=0. 7322，deviates from exp data by 2.09。 The analogue in thermodynamicsL1L2L3L1”L2”L3”Mat

14、ter appears similarly at critical pointmanchickenman/100chicken/5child100.50.10.1young2010.20.2adult4020.40.4old6030.60.6Gases with same Tr and Pr keep the same deviation degree from ideal gas.The Nelson and Obett correlation graphp362.981.05p3619.21.1511.961.2014.321.10p37At low pressures, Pr=01, 3

15、0 gases are used，erro1%At moderate pressures, Pr=110, 30 gases are used， erro2.5%, except for H2, He, NH3, F2, CH4At high pressures, Pr=1040, less data，Tr=13.5, Pr=1020, erro2.5，Z1 at any range of P. Vreal gasVideal gas (under the same T and P) , the real gas is more difficult to compress (consuming

16、 more work)。(2)When Tr2.5 and Pr is low，there is a minimum value of Z, Z1Vreal gas10， all gases deviate ideal gas to a remarkable extent。(6) When Pr0， Z1 ( all gases can be regarded as ideal gas).The two parameter correlation gives rough predictions For most gases, Zc=0.250.31。So, roughly: Suitable

17、for symmetric molecules such as argon(氬)、krypton (氪)、xenon(氙). For nonsymmetrical and polar molecules, remarkable deviations will be expected。 2.4.4 Z graph with Tr,Pr and as parameters 三參數普遍化壓縮因子圖: acentric factor, defined with reference of vapor pressure. We set :At critical point: Tr=Pr=1, then a

18、=blgPr is linear function of 1/Tr ，with a slope of “-a”. It seems available for all the gases。But “a” is deferent from gas to gas:The acentric factor See app. B1, P654- Represents the deviation degree from spherical moleculesPitzers correlationThe Lee-Kesler generalized Z0 graph (Pr1.0)0.865Z0 graph

19、 (Pr 1.0)2.5050.57The Z1 graph (Pr1.0)0.036The generalized Z1 graph (Pr 1.0)2.5050.16The Lee-Kesler Generalized Correlation Tables App. D , P263-2702.4.5 The generalized form of virial eqn 普遍化維里方程Where，(BPc)/(RTc) is dimensionless，the reduced 2nd Virial coefficient, a function of T only. Pitzer prop

20、osed： B0、B1 are functions of Tr too. Pitzer suggests: Available only under low to moderate pressure The suitable application range for Virial eqn with the 2nd coefficient Further modification by Tsonopoulos :0.2110.833Ex. 2.6Solution：From appendix B, Tc=425.2K, Pc=3.8MPa, =0.1931) By ideal gas law:2

21、) By generalized Z correlation: 3) By generalized Virial-coefficient correlation: n-C4, T=510K, P=2.5MPa, Vm=? Vm,exp=1.48m3/kmolBoth 2) and 3) give a result quite close to the experimental value Ex.NH3, m=0.5kg, V=0.03m3, T=65+273.16=338.16K, P=? (Pexp=2.382 Mpa)Solution：a . By ideal gas eqn:b . By R-K eqn: from appendix: Tc=405.6K, Pc=11.28 MPac) With a generalized correlation: sine Pr is small, the generalized virial eqn is adopted . Here, Tc=405.6K, Pc=11.28 Mpa, =0.25Tr=338.16/405.6=0.833, Pr=2.387/11.28=0.211Erro: a) 15.6% , b) 0.04%, c) 0.16%2.5 Volumetric Properties of Liqu