Lecture 11 Critical point and real gases

Phase diagrams of water and CO2 H2O CO2 Phase transition from solid to gas is called sublimation If T >Tcritlical and p > pcritical fluid is called a supercritical fluid

Specific volumes of saturated water (v’) and steam/vapor (v’’) TC = 374.15 oC and pC = 221.2 bar At the critical point, molar/specific volumes (or densities) of the saturated liquid and vapor are the same.

Isotherms of water in p,v-diagram Liduid Gas

Critical values for some gases pC. bar tC. oC vC. dm3/kg He 2.29 -267.9 14.49 H2 12.96 -239.9 32.26 O2 50.35 -118.8 2.326 N2 33.98 -146.9 3.215 Air 37.60 -140.6 2.830 H2O 221.1 374.2 3.18 CO2 73.6 31.0 2.156 NH3 (ammoniac) 112.95 132.4 4.255 CH4 (methane) 46.30 -82.5 6.173

A supercritical power plant A non-supercitical power plant pfeed water < pC A supercitical power plant pfeed water > pC To the turbine To the turbine Superheated steam 190bar. 540oC Supercritical steam 240bar. 540oC Liquid water 200bar. 202oC Liquid water 250bar. 202oC From the feed water tank From the feed water tank Suprcritical boilers are once-through boilers with no steam drums, for example a Benson boiler

Specific heat capacities at the critical point t. oC p. bar cp’. specific heat of saturated water. kJ/kgK cp’’. specific heat of saturated steam/vapor. kJ/kgK 10 0.012270 4.193 1.860 50 0.12335 4.181 1.899 100 1.013 4.216 2.028 150 4.760 4.310 2.314 200 15.549 4.497 2.843 250 39.776 4.867 3.772 300 85.927 5.762 5.863 330 128.63 7.219 9.361 350 165.35 10.11 17.15 360 186.75 14.58 25.12 370 210.54 43.17 76.92 374.15 221.20 

Ideal gas versus real gases Molecules are point masses. There is no interaction between molecules. The model is usually valid when densities are low. Real gas Molecules take a finite volume => the volume cannot be smaller than the total molecular volume of the gas. Molecules interact with each other. Gases do not obey the ideal gas equation of state when gas temperatures and pressures begin to ”approach” the critical point.

Some equations of state for real gases wan der Waals equation a and b are constants which have different values for a given gas. Redlich-Kwong equation a and b are constants which have different values for a given gas. Other real gas equations - Beattie Bridgeman (is used for water) - The virial equation of state

Calculating the constants a and b wan der Waals equation Redlich-Kwong equation TC critical temperature and pC critical pressure

Calculating the enthalpy change using van der Waals equation Total differential of the internal energy => => => the enthalpy becomes h = u +pvm [J/mol] cv is the specific heat capacity at constant volume [J/molK] The dependence of the enthalpy on the pressure

Example 1: van der Waals vs. Ideal gas Evaluate the accuracy of the van der Waals and ideal gas equations for water vapor at 300oC. TC = 374.2oC pC = 221.1bar wan der Waals equation Ideal gas equation

Example 1: van der Waals vs. Ideal gas results bar vm van der Waals m3/mol vm. ideal vm real. m3/mol 0.40 0.120000 0.120034 0.118919 1.33 0.035834 0.035868 0.035684 2.65 0.017958 0.017992 0.017851 5.28 0.008990 0.009023 0.008901 10.52 0.004497 0.004531 0.004407 14.42 0.003271 0.003305 0.003182 19.59 0.002399 0.002433 0.002308 24.72 0.001894 0.001928 0.001802 29.81 0.001599 0.001470 34.86 0.001333 0.001367 0.001237 38.63 0.001200 0.001234 0.001102 43.62 0.001059 0.001092 0.000958 47.34 0.000973 0.001007 0.000870 52.27 0.000878 0.000912 0.000773 55.94 0.000818 0.000852 0.000711 59.59 0.000766 0.000800 0.000656 64.43 0.000706 0.000740 0.000593 69.23 0.000654 0.000688 0.000538 72.81 0.000621 0.000502 75.19 0.000600 0.000634 0.000479

Principle of the corresponding state and the compression factor The compression factor Z is defined for all gases as follows: where Z(pr . Tr) is a universal function that can be used for all gases. The accuracy of Z should be Z  6%

Nelson-Obert Generalized Compressibility Chart for Gases If Z  1 => gas is an ideal gas If Z differs from 1 => gas is a real gas

Example 2: Gas density Calculate the density of methane at 6bar and 17oC using the generalized compressibility chart and the ideal gas equation. For methane TC = 191.1K and pC = 46.4bar From the Generalized Compressibility Chart Z  0.99 => The ideal gas equation

Example 2: Gas density

Example 2: Gas density At 100bar and -23oC. the corresponding densities for CH4 would be The Generalized Compressibility Chart => The ideal gas equation

Example 3: Throttling valve What is the temperture of N2 after the valve? Throttling valve N2 . 100bar. -10oC N2 . 70bar. t2? The energy balance over the valve w1  w2 => h1 = h2

Example 3: Throttling valve The change of the enthalpy where and dh = 0 => where  is the volumetric expansion coefficient

Example 3: Throttling valve For N2 PC = 33.6 bar and TC = 126.25K => Z1  0.97 and Note: dfg = f’g +fg’

Example 3: Throttling valve The term Z(T.p)/ Z(T) may be approximated from the Nelson-Obert chart as follows => After integration

Example 3: Throttling valve Tr1 Z Tr1’’ Z1 and Tr1 represent the values before the valve

Example 3: Throttling valve Numerical values on the basis of the compressibility chart cpm (70bar. 263.15K)  33.7J/molK => => t2 = -18.5oC . for an ideal gas T = 0oC