Exercise 9 solution text

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Prof. Gianpiero Groppi – Exercises – Fundamentals of Chemical Processes – A.A. 2023-2024 1 EXERCISE 9 Estimation of the bubble and dew temperatures of a n- butane/n-hexane mixture A mixture composed of n-butane and n-hexane (1:1 v/v) is kept at 4.85 bar. Estimate its bubble and dew temperature in the following cases: • Ideal gases and ideal mixture • Real gases and ideal mixture • Real gases and real mixture Assume the RKS equation of state. DATA ANTOINE COEFFICIENTS TC [K] PC [bar] ω A B C n-C4H10 13.6608 2154.700 238.789 425.10 37.96 0.200 n-C6H14 13.8193 2696.040 224.317 507.60 30.25 0.301 Antoine: ln𝑝𝑖 0=𝐴𝑖− 𝐵𝑖 𝑇+𝐶𝑖 with P in kPa, T in °C RKS parameters: 𝑎=0.42748⋅𝛼⋅(𝑅𝑔𝑎𝑠⋅𝑇𝐶) 𝑃𝐶 2 𝑏=0.08664⋅𝑅𝑔𝑎𝑠⋅𝑇𝐶 𝑃𝐶 𝛼=(1+𝑆⋅(1−√𝑇𝑅)) 2 𝑆=0.48+1.574⋅𝜔−0.176⋅𝜔2 𝐴= 𝑎⋅𝑃 (𝑅𝑔𝑎𝑠⋅𝑇) 2 𝐵= 𝑏⋅𝑃 𝑅𝑔𝑎𝑠⋅𝑇 Fugacity coefficient (Z indicates the root of the cubic equation): ln𝜑(𝑇,𝑃)=𝑍−1−𝐴 𝐵⋅ln(𝑍+𝐵 𝑍 )−ln(𝑍−𝐵) Prof. Gianpiero Groppi – Exercises – Fundamentals of Chemical Processes – A.A. 2023-2024 2 RKS parameters for mixtures and mixing rules: 𝑎𝑚𝑖𝑥=(∑𝑥𝑖.√𝑎𝑖 𝑁𝐶 𝑖 ) 2 𝑏𝑚𝑖𝑥=∑𝑥𝑖.𝑏𝑖 𝑁𝐶 𝑖 𝐴𝑚𝑖𝑥= 𝑎𝑚𝑖𝑥⋅𝑃 (𝑅𝑔𝑎𝑠⋅𝑇) 2 𝐵𝑚𝑖𝑥=𝑏𝑚𝑖𝑥⋅𝑃 𝑅𝑔𝑎𝑠⋅𝑇 Fugacity coefficient: ln𝜑̂𝑖 ∝(𝑇,𝑃,𝑥∝)= 𝐵𝑖 𝐵𝑚𝑖𝑥 .(𝑍−1)+𝐴𝑚𝑖𝑥 𝐵𝑚𝑖𝑥 .( 𝐵𝑖 𝐵𝑚𝑖𝑥 −2.√ 𝐴𝑖 𝐴𝑚𝑖𝑥 ).ln(𝑍+𝐵𝑚𝑖𝑥 𝑍 )−ln(𝑧−𝐵𝑚𝑖𝑥) Solving procedure for the cubic equation: 𝑍3+𝛼⋅𝑍2+𝛽⋅𝑍+𝛾=0 𝛼=−1 𝛽=𝐴−𝐵−𝐵2 𝛾=−𝐴⋅𝐵 𝑝=𝛽−𝛼2 3 𝑞=2𝛼3 27 −𝛼⋅𝛽 3 +𝛾 𝐷=𝑞2 4 +𝑝3 27 If D>0, only 1 real solution is found: 𝑍=(−𝑞 2+√𝐷) 1 3 +(−𝑞 2−√𝐷) 1 3 −𝛼 3 If D = 0, 3 real solutions are found (2 of which are identical): 𝑍1=−2⋅(−𝑞 2) 1 3 −𝛼 3 𝑍2=𝑍3=(−𝑞 2) 1 3 −𝛼 3 If D< 0, 3 distinct real solutions are found: 𝑍1=2⋅𝑟 1 3𝑐𝑜𝑠(𝜃 3)−𝛼 3 𝑍2=2⋅𝑟 1 3𝑐𝑜𝑠(2𝜋+𝜃 3 )−𝛼 3 𝑍3=2⋅𝑟 1 3𝑐𝑜𝑠(4𝜋+𝜃 3 )−𝛼 3 𝑟=√−𝑝3 27 𝑐𝑜𝑠(𝜃)=− 𝑞 2𝑟 Prof. Gianpiero Groppi – Exercises – Fundamentals of Chemical Processes – A.A. 2023-2024 3 SOLUTION This problem…