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"cell_type": "heading",
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"source": [
"\n",
"Chapter 13 : Fugacity of a Component in a Mixture by Equations of State"
]
},
{
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"level": 3,
"metadata": {},
"source": [
"Example 13.2 Page Number : 433"
]
},
{
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"input": [
" \n",
"import math \n",
"\n",
"\n",
"# Variables\n",
"T = 310.93;\t\t\t#[K] - Temperature\n",
"P = 2.76*10**(6);\t\t\t#[Pa] - Pressure\n",
"y1 = 0.8942;\t\t\t#[mol] - mole fraction of component 1\n",
"y2 = 1 - y1;\t\t\t#[mol] - mole fraction of component 2\n",
"R=8.314;\t\t\t#[J/mol*K] - Universal gas constant\n",
"\n",
"#For component 1 (methane)\n",
"Tc_1 = 190.6;\t\t\t#[K] - Critical temperature\n",
"Pc_1 = 45.99*10**(5);\t\t\t#[N/m**(2)] - Critical pressure\n",
"Vc_1 = 98.6;\t\t\t#[cm**(3)/mol] - Critical molar volume\n",
"Zc_1 = 0.286;\t\t\t# - Critical compressibility factor\n",
"w_1 = 0.012;\t\t\t# - Critical acentric factor\n",
"#Similarly for component 2 (n-Butane)\n",
"Tc_2 = 425.1;\t\t\t#[K]\n",
"Pc_2 = 37.96*10**(5);\t\t\t#[N/m**(2)]\n",
"Vc_2 = 255;\t\t\t#[cm**(3)/mol]\n",
"Zc_2=0.274;\n",
"w_2=0.2;\n",
"\n",
"# Calculations\n",
"#For component 1\n",
"Tr_1 = T/Tc_1;\t\t\t#Reduced temperature\n",
"#At reduced temperature\n",
"B1_0 = 0.083-(0.422/(Tr_1)**(1.6));\n",
"B1_1 = 0.139-(0.172/(Tr_1)**(4.2));\n",
"#We know,(B*Pc)/(R*Tc) = B_0+(w*B_1)\n",
"B_11 = ((B1_0+(w_1*B1_1))*(R*Tc_1))/Pc_1;\t\t\t#[m**(3)/mol]\n",
"\n",
"#Similarly for component 2\n",
"Tr_2 = T/Tc_2;\t\t\t#Reduced temperature\n",
"#At reduced temperature Tr_2,\n",
"B2_0 = 0.083-(0.422/(Tr_2)**(1.6));\n",
"B2_1 = 0.139-(0.172/(Tr_2)**(4.2));\n",
"B_22 = ((B2_0+(w_2*B2_1))*(R*Tc_2))/Pc_2;\t\t\t#[m**(3)/mol]\n",
"\n",
"#For cross coeffcient\n",
"Tc_12 = (Tc_1*Tc_2)**(1./2);\t\t\t#[K]\n",
"w_12 = (w_1+w_2)/2;\n",
"Zc_12 = (Zc_1+Zc_2)/2;\n",
"Vc_12 = (((Vc_1)**(1./3)+(Vc_2)**(1./3))/2)**(3);\t\t\t#[cm**(3)/mol]\n",
"Vc_12 = Vc_12*10**(-6);\t\t\t#[m**(3)/mol]\n",
"Pc_12 = (Zc_12*R*Tc_12)/Vc_12;\t\t\t#[N/m**(2)]\n",
"\n",
"# Variables, Z = 1 + (B*P)/(R*T)\n",
"#Now we have,(B_12*Pc_12)/(R*Tc_12) = B_0 + (w_12*B_1)\n",
"#where B_0 and B_1 are to be evaluated at Tr_12\n",
"Tr_12 = T/Tc_12;\n",
"#At reduced temperature Tr_12\n",
"B_0 = 0.083-(0.422/(Tr_12)**(1.6));\n",
"B_1 = 0.139-(0.172/(Tr_12)**(4.2));\n",
"B_12 = ((B_0+(w_12*B_1))*R*Tc_12)/Pc_12;\t\t\t#[m**(3)/mol]\n",
"#For the mixture\n",
"B = y1**(2)*B_11+2*y1*y2*B_12+y2**(2)*B_22;\t\t\t#[m**(3)/mol]\n",
"\n",
"#Now del_12 can be calculated as,\n",
"del_12 = 2*B_12 - B_11 - B_22;\t\t\t#[m**(3)/mol]\n",
"\n",
"#We have the relation, math.log(phi_1) = (P/(R*T))*(B_11 + y2**(2)*del_12), therefore\n",
"phi_1 = math.exp((P/(R*T))*(B_11 + y2**(2)*del_12));\n",
"#Similarly for component 2\n",
"phi_2 = math.exp((P/(R*T))*(B_22 + y1**(2)*del_12));\n",
"\n",
"# Results\n",
"print \" The value of fugacity coefficient of component 1 phi_1) is %f\"%(phi_1);\n",
"print \" The value of fugacity coefficient of component 2 phi_2) is %f\"%(phi_2);\n",
"\n",
"#Finally fugacity coefficient of the mixture is given by\n",
"#math.log(phi) = y1*math.log(phi_1) + y2*math.log(phi_2);\n",
"phi = math.exp(y1*math.log(phi_1) + y2*math.log(phi_2));\n",
"\n",
"print \" The value of fugacity coefficient of the mixture phi) is %f \"%(phi);\n",
"#The fugacity coefficient of the mixture can also be obtained using\n",
"#math.log(phi) = (B*P)/(R*T)\n",
"\n"
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" The value of fugacity coefficient of component 1 phi_1) is 0.965152\n",
" The value of fugacity coefficient of component 2 phi_2) is 0.675374\n",
" The value of fugacity coefficient of the mixture phi) is 0.929376 \n"
]
}
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"cell_type": "heading",
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"source": [
"Example 13.7 Page Number : 447"
]
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" \n",
"from scipy.optimize import fsolve \n",
"import math \n",
"\n",
"# Variables\n",
"T = 460.;\t\t\t#[K] - Temperature\n",
"P = 40.*10**(5);\t\t\t#[Pa] - Pressure\n",
"R=8.314;\t\t\t#[J/mol*K] - Universal gas constant\n",
"# component 1 = nitrogen\n",
"# component 2 = n-Butane\n",
"y1 = 0.4974;\t\t\t# Mole percent of nitrogen\n",
"y2 = 0.5026;\t\t\t# Mole percent of n-Butane\n",
"Tc_nit = 126.2;\t\t\t#[K]\n",
"Pc_nit = 34.00*10**(5);\t\t\t#[Pa]\n",
"Tc_but = 425.1;\t\t\t#[K]\n",
"Pc_but = 37.96*10**(5);\t\t\t#[Pa]\n",
"\n",
"# (1). van der Walls equation of state\n",
"\n",
"# The fugacity coefficient of component 1 in a binary mixture following van der Walls equation of state is given by,\n",
"# math.log(phi_1) = b_1/(V-b) - math.log(Z-B) -2*(y1*a_11 + y2*a_12)/(R*T*V)\n",
"# and for component 2 is given by,\n",
"# math.log(phi_2) = b_2/(V-b) - math.log(Z-B) -2*(y1*a_12 + y2*a_22)/(R*T*V)\n",
"# Where B = (P*b)/(R*T)\n",
"\n",
"# Calculations\n",
"# For componenet 1 (nitrogen)\n",
"a_1 = (27*R**(2)*Tc_nit**(2))/(64*Pc_nit);\t\t\t#[Pa-m**(6)/mol**(2)]\n",
"b_1 = (R*Tc_nit)/(8*Pc_nit);\t\t\t#[m**(3)/mol]\n",
"\n",
"# Similarly for componenet 2 (n-Butane)\n",
"a_2 = (27*R**(2)*Tc_but**(2))/(64*Pc_but);\t\t\t#[Pa-m**(6)/mol**(2)]\n",
"b_2 = (R*Tc_but)/(8*Pc_but);\t\t\t#[m**(3)/mol]\n",
"\n",
"# Here\n",
"a_11 = a_1;\n",
"a_22 = a_2;\n",
"# For cross coefficient\n",
"a_12 = (a_1*a_2)**(1./2);\t\t\t#[Pa-m**(6)/mol**(2)]\n",
"\n",
"# For the mixture \n",
"a = y1**(2)*a_11 + y2**(2)*a_22 + 2*y1*y2*a_12;\t\t\t#[Pa-m**(6)/mol**(2)]\n",
"b = y1*b_1 + y2*b_2;\t\t\t#[m**(3)/mol]\n",
"\n",
"# The cubic form of the van der Walls equation of state is given by,\n",
"# V**(3) - (b+(R*T)/P)*V**(2) + (a/P)*V - (a*b)/P = 0\n",
"# Substituting the value and solving for V, we get\n",
"# Solving the cubic equation\n",
"def f(V): \n",
" return V**(3)-(b+(R*T)/P)*V**(2)+(a/P)*V-(a*b)/P\n",
"V_1=fsolve(f,-1)\n",
"V_2=fsolve(f,0)\n",
"V_3=fsolve(f,1)\n",
"# The molar volume V = V_3, the other two roots are imaginary\n",
"V = V_3;\t\t\t#[m**(3)/mol]\n",
"\n",
"# The comprssibility factor of the mixture is \n",
"Z = (P*V)/(R*T);\n",
"# And B can also be calculated as\n",
"B = (P*b)/(R*T);\n",
"\n",
"# The fugacity coefficient of component 1 in the mixture is\n",
"phi_1 = math.exp(b_1/(V-b) - math.log(Z-B) -2*(y1*a_11 + y2*a_12)/(R*T*V));\n",
"# Similarly fugacity coefficient of component 2 in the mixture is \n",
"phi_2 = math.exp(b_2/(V-b) - math.log(Z-B) -2*(y1*a_12 + y2*a_22)/(R*T*V));\n",
"\n",
"# The fugacity coefficient of the mixture is given by,\n",
"# math.log(phi) = y1*math.log(phi_1) + y2*math.log(phi_2)\n",
"phi = math.exp(y1*math.log(phi_1) + y2*math.log(phi_2));\n",
"\n",
"# Also the fugacity coefficient of the mixture following van der Walls equation of state is given by,\n",
"# math.log(phi) = b/(V-b) - math.log(Z-B) -2*a/(R*T*V)\n",
"phi_dash = math.exp(b/(V-b) - math.log(Z-B) -2*a/(R*T*V));\n",
"# The result is same as obtained above\n",
"\n",
"# Results\n",
"print \" 1van der Walls equation of state\";\n",
"print \" The value of fugacity coefficient of component 1 nitrogen) is %f\"%(phi_1);\n",
"print \" The value of fugacity coefficient of component 2 n-butane) is %f\"%(phi_2);\n",
"print \" The value of fugacity coefficient of the mixture is %f\"%(phi);\n",
"print \" Also the fugacity coefficient of the mixture from van der Walls equation of state is %f which is same as above)\"%(phi_dash);\n",
"\n",
"# (2). Redlich-Kwong equation of state\n",
"\n",
"# For component 1,\n",
"a_1_prime = (0.42748*R**(2)*Tc_nit**(2.5))/Pc_nit;\t\t\t#[Pa-m**(6)/mol**(2)]\n",
"b_1_prime = (0.08664*R*Tc_nit)/Pc_nit;\t\t\t#[m**(3)/mol]\n",
"\n",
"#similarly for component 2,\n",
"a_2_prime = (0.42748*R**(2)*Tc_but**(2.5))/Pc_but;\t\t\t#[Pa-m**(6)/mol**(2)]\n",
"b_2_prime = (0.08664*R*Tc_but)/Pc_but;\t\t\t#[m**(3)/mol]\n",
"\n",
"# For cross coefficient\n",
"a_12_prime = (a_1_prime*a_2_prime)**(1./2);\t\t\t#[Pa-m**(6)/mol**(2)]\n",
"# For the mixture\n",
"a_prime = y1**(2)*a_1_prime + y2**(2)*a_2_prime +2*y1*y2*a_12_prime;\t\t\t#[Pa-m**(6)/mol**(2)]\n",
"b_prime = y1*b_1_prime +y2*b_2_prime;\t\t\t#[m**(3)/mol]\n",
"\n",
"\n",
"#The cubic form of Redlich Kwong equation of state is given by,\n",
"# V**(3)-((R*T)/P)*V**(2)-((b**(2))+((b*R*T)/P)-(a/(T**(1/2)*P))*V-(a*b)/(T**(1/2)*P)=0\n",
"# Solving the cubic equation\n",
"def f1(V): \n",
" return V**(3)-((R*T)/P)*V**(2)-((b_prime**(2))+((b_prime*R*T)/P)-(a_prime/(T**(1./2)*P)))*V-(a_prime*b_prime)/(T**(1./2)*P)\n",
"V_4=fsolve(f1,1)\n",
"V_5=fsolve(f1,0)\n",
"V_6=fsolve(f1,-1)\n",
"# The molar volume V = V_4, the other two roots are imaginary\n",
"V_prime = V_4;\t\t\t#[m**(3)/mol]\n",
"\n",
"# The compressibility factor of the mixture is\n",
"Z_prime = (P*V_prime)/(R*T);\n",
"# And B can also be calculated as\n",
"B_prime = (P*b_prime)/(R*T);\n",
"\n",
"# The fugacity coefficient of component 1 in the binary mixture is given by\n",
"# math.log(phi_1) = b_1/(V-b) - math.log(Z-B) + ((a*b_1)/((b**(2)*R*T**(3/2))))*(math.log((V+b)/V)-(b/(V+b)))-(2*(y1*a_1+y2*a_12)/(R*T**(3/2)b))*(math.log(V+b)/b)\n",
"\n",
"phi_1_prime = math.exp((b_1_prime/(V_prime-b_prime))-math.log(Z_prime-B_prime)+((a_prime*b_1_prime)/((b_prime**(2))*R*(T**(3./2))))*(math.log((V_prime+b_prime)/V_prime)-(b_prime/(V_prime+b_prime)))-(2*(y1*a_1_prime+y2*a_12_prime)/(R*(T**(3./2))*b_prime))*(math.log((V_prime+b_prime)/V_prime)));\n",
"\n",
"\n",
"# Similarly fugacity coefficient of component 2 in the mixture is \n",
"phi_2_prime = math.exp((b_2_prime/(V_prime-b_prime))-math.log(Z_prime-B_prime)+((a_prime*b_2_prime)/((b_prime**(2))*R*(T**(3./2))))*(math.log((V_prime+b_prime)/V_prime)-(b_prime/(V_prime+b_prime)))-(2*(y1*a_12_prime+y2*a_2_prime)/(R*(T**(3./2))*b_prime))*(math.log((V_prime+b_prime)/V_prime)));\n",
"\n",
"# The fugacity coefficient of the mixture is given by,\n",
"# math.log(phi) = y1*math.log(phi_1) + y2*math.log(phi_2)\n",
"phi_prime = math.exp(y1*math.log(phi_1_prime) + y2*math.log(phi_2_prime));\n",
"\n",
"# Also the fugacity coefficient for the mixture following Redlich-kwong equation of state is also given by\n",
"# math.log(phi) = b/(V-b) - math.log(Z-B) - (a/(R*T**(3/2)))*(1/(V+b)+(1/b)*math.log((V+b)/V))\n",
"phi_prime_dash = math.exp(b_prime/(V_prime-b_prime) - math.log(Z_prime-B_prime) - (a_prime/(R*T**(3./2)))*(1/(V_prime+b_prime)+(1./b_prime)*math.log((V_prime+b_prime)/ V_prime)));\n",
"\n",
"print \" \\nRedlich-Kwong equation of state\";\n",
"print \" The value of fugacity coefficient of component 1 nitrogen) is %f\"%(phi_1_prime);\n",
"print \" The value of fugacity coefficient of component 2 n-butane) is %f\"%(phi_2_prime);\n",
"print \" The value of fugacity coefficient of the mixture is %f\"%(phi_prime);\n",
"print \" Also the fugacity coefficient for the mixture from Redlich-kwong equation of state is %f which is same as above)\"%(phi_prime_dash);\n"
],
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"text": [
" 1van der Walls equation of state\n",
" The value of fugacity coefficient of component 1 nitrogen) is 1.057500\n",
" The value of fugacity coefficient of component 2 n-butane) is 0.801865\n",
" The value of fugacity coefficient of the mixture is 0.920192\n",
" Also the fugacity coefficient of the mixture from van der Walls equation of state is 0.920192 which is same as above)\n",
" \n",
"Redlich-Kwong equation of state\n",
" The value of fugacity coefficient of component 1 nitrogen) is 1.071129\n",
" The value of fugacity coefficient of component 2 n-butane) is 0.793063\n",
" The value of fugacity coefficient of the mixture is 0.920948\n",
" Also the fugacity coefficient for the mixture from Redlich-kwong equation of state is 0.920948 which is same as above)\n"
]
}
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