Thermodynamics - Comparison of Gas Laws

162 days ago by DorkeyDear

# Variables P = var('P'); # Pressure (Pa) SV = var('v', latex_name='\overline{v}'); # Specific Volume (m^3 / kg) T = var('T'); # Temperature (K) # Constants MW = 0.04401; # Molar Weight (kg / mol) R = 188.9; # Specific Gas Constant (J / kg / K) R = R / MW; # Molar Specific Gas Constant (J / mol / K) a = 0.364; #0.07132 # (J * m^3 / mol^2) b = 4.267E-5; # (m^3 / mol) a0 = 0.07132; b0 = 0.07235; c0 = 660000; A0 = 507.2836; B0 = 0.10476; A = A0 * (1 - a0 / v); B = B0 * (1 - b0 / v); # none to self: answers may be in molar units, not per mass; need to double check # note to self: need to check what material this is for, so reasonable ranges of P and T for where these laws are applicable I can use; currently P from 0.5 - 100 atm (only 10 atm in plots), and T from 200 to 400 K (-73.15 to 126.85 ºC) 
       
# Gas Laws # Ideal Gas Law IGL = P * SV == R * T; show(IGL); # van der Waals Equation of State VDW = (P + a / SV^2) * (v - b) == R * T; show(VDW); # Beattie-Bridgeman Equation of State BB = P == R * T / v^2 * (1 - c0 / (v * T^3)) * (v + B) - A / v^2; show(BB); 
        
\newcommand{\Bold}[1]{\mathbf{#1}}P \overline{v} = 4292.20631674619 \, T


\newcommand{\Bold}[1]{\mathbf{#1}}{\left(\overline{v} - 0.0000426700000000000\right)} {\left(P + \frac{0.364000000000000}{\overline{v}^{2}}\right)} = 4292.20631674619 \, T


\newcommand{\Bold}[1]{\mathbf{#1}}P = -\frac{4292.20631674619 \, {\left(\frac{660000}{T^{3} \overline{v}} - 1\right)} {\left(\overline{v} - \frac{0.00757938600000000}{\overline{v}} + 0.104760000000000\right)} T}{\overline{v}^{2}} - \frac{-\frac{36.1794663520000}{\overline{v}} + 507.283600000000}{\overline{v}^{2}}
\newcommand{\Bold}[1]{\mathbf{#1}}P \overline{v} = 4292.20631674619 \, T


\newcommand{\Bold}[1]{\mathbf{#1}}{\left(\overline{v} - 0.0000426700000000000\right)} {\left(P + \frac{0.364000000000000}{\overline{v}^{2}}\right)} = 4292.20631674619 \, T


\newcommand{\Bold}[1]{\mathbf{#1}}P = -\frac{4292.20631674619 \, {\left(\frac{660000}{T^{3} \overline{v}} - 1\right)} {\left(\overline{v} - \frac{0.00757938600000000}{\overline{v}} + 0.104760000000000\right)} T}{\overline{v}^{2}} - \frac{-\frac{36.1794663520000}{\overline{v}} + 507.283600000000}{\overline{v}^{2}}
# Preparation for plotting pIGL = implicit_plot3d(IGL, (P, 0.5*100E3, 10*100E3), (T, 200, 400), (SV, 0, 35), color='red' ); pVDW = implicit_plot3d(VDW, (P, 0.5*100E3, 10*100E3), (T, 200, 400), (SV, 0, 35), color='green'); pBB = implicit_plot3d(BB, (P, 0.5*100E3, 10*100E3), (T, 200, 400), (SV, 0, 35), color='blue' ); 
       
# Plotting Ideal Gas Law show(pIGL); 
       
# Plotting van der Waals Equation of State show(pVDW); 
       
# Plotting Beattie-Bridgeman Equation of State show(pBB); 
       
# Combining all plots for comparison purposes show(pIGL + pVDW + pBB); 
       
import numpy # Pressure and temperature sweeps Parray = numpy.linspace(0.5 * 100E3, 100 * 100E3, 5) Tarray = numpy.linspace(200, 400, 5) # Table header Table = [['Pressure\n(MPa)', 'Temperature\n(ºC)', 'Specific Volume\nIdeal Gas Law\n(kg/m^3)', 'Specific Volume\nvan der Waals\n(kg/m^3)', 'Specific Volume\nBeattie-Bridgeman\n(kg/m^3)']] # Table generating for Pelem in Parray: for Telem in Tarray: SV_IGL = IGL.substitute({P: Pelem, T: Telem}).find_root(0, 35) SV_VDW = VDW.substitute({P: Pelem, T: Telem}).find_root(0, 35) SV_BB = BB.substitute({P: Pelem, T: Telem}).find_root(0, 35) Table += [[round(Pelem / 1000000.0, 6), round(Telem - 273.15, 6), round(SV_IGL, 6), round(SV_VDW, 6), round(SV_BB, 6)]] # Display the table html.table(Table, header=True) 
        






Pressure
(MPa)
Temperature
(ºC)
Specific Volume
Ideal Gas Law
(kg/m^3)
Specific Volume
van der Waals
(kg/m^3)
Specific Volume
Beattie-Bridgeman
(kg/m^3)




0.05
-73.15
17.168825
17.168868
17.18953




0.05
-23.15
21.461032
21.461074
21.522348




0.05
26.85
25.753238
25.75328
25.832524




0.05
76.85
30.045444
30.045487
30.133908




0.05
126.85
34.337651
34.337693
34.431296




2.5375
-73.15
0.338302
0.338344
0.31268




2.5375
-23.15
0.422877
0.42292
0.457459




2.5375
26.85
0.507453
0.507495
0.563771




2.5375
76.85
0.592028
0.592071
0.659694




2.5375
126.85
0.676604
0.676646
0.751175




5.025
-73.15
0.170834
0.170876
0.045803




5.025
-23.15
0.213543
0.213585
0.228277




5.025
26.85
0.256251
0.256294
0.297611




5.025
76.85
0.29896
0.299002
0.353143




5.025
126.85
0.341668
0.341711
0.403653




7.5125
-73.15
0.114268
0.114311
0.044676




7.5125
-23.15
0.142835
0.142878
0.036938




7.5125
26.85
0.171403
0.171445
0.200552




7.5125
76.85
0.19997
0.200012
0.243762




7.5125
126.85
0.228537
0.228579
0.280921




10.0
-73.15
0.085844
0.085886
0.043747




10.0
-23.15
0.107305
0.107347
0.036117




10.0
26.85
0.128766
0.128809
0.145794




10.0
76.85
0.150227
0.15027
0.185049




10.0
126.85
0.171688
0.171731
0.216063












Pressure
(MPa)
Temperature
(ºC)
Specific Volume
Ideal Gas Law
(kg/m^3)
Specific Volume
van der Waals
(kg/m^3)
Specific Volume
Beattie-Bridgeman
(kg/m^3)




0.05
-73.15
17.168825
17.168868
17.18953




0.05
-23.15
21.461032
21.461074
21.522348




0.05
26.85
25.753238
25.75328
25.832524




0.05
76.85
30.045444
30.045487
30.133908




0.05
126.85
34.337651
34.337693
34.431296




2.5375
-73.15
0.338302
0.338344
0.31268




2.5375
-23.15
0.422877
0.42292
0.457459




2.5375
26.85
0.507453
0.507495
0.563771




2.5375
76.85
0.592028
0.592071
0.659694




2.5375
126.85
0.676604
0.676646
0.751175




5.025
-73.15
0.170834
0.170876
0.045803




5.025
-23.15
0.213543
0.213585
0.228277




5.025
26.85
0.256251
0.256294
0.297611




5.025
76.85
0.29896
0.299002
0.353143




5.025
126.85
0.341668
0.341711
0.403653




7.5125
-73.15
0.114268
0.114311
0.044676




7.5125
-23.15
0.142835
0.142878
0.036938




7.5125
26.85
0.171403
0.171445
0.200552




7.5125
76.85
0.19997
0.200012
0.243762




7.5125
126.85
0.228537
0.228579
0.280921




10.0
-73.15
0.085844
0.085886
0.043747




10.0
-23.15
0.107305
0.107347
0.036117




10.0
26.85
0.128766
0.128809
0.145794




10.0
76.85
0.150227
0.15027
0.185049




10.0
126.85
0.171688
0.171731
0.216063