## Calculating a bubble point pressure of a mixture

| categories: nonlinear algebra | tags: thermodynamics | View Comments

We previously learned to read a datafile containing lots of Antoine coefficients into a database, and use the coefficients to estimate vapor pressure of a single compound. Here we use those coefficents to compute a bubble point pressure of a mixture.

The bubble point is the temperature at which the sum of the component vapor pressures is equal to the the total pressure. This is where a bubble of vapor will first start forming, and the mixture starts to boil.

Consider an equimolar mixture of benzene, toluene, chloroform, acetone and methanol. Compute the bubble point at 760 mmHg, and the gas phase composition. The gas phase composition is given by: $$y_i = x_i*P_i/P_T$$.

import numpy as np
from scipy.optimize import fsolve

dtype=[('id', np.int),
('formula', 'S8'),
('name', 'S28'),
('A', np.float),
('B', np.float),
('C', np.float),
('Tmin', np.float),
('Tmax', np.float),
('??', 'S4'),
('?', 'S4')],
skiprows=7)

compounds = ['benzene', 'toluene', 'chloroform', 'acetone', 'methanol']

# extract the data we want
A = np.array([data[data['name'] == x]['A'][0] for x in compounds])
B = np.array([data[data['name'] == x]['B'][0] for x in compounds])
C = np.array([data[data['name'] == x]['C'][0] for x in compounds])
Tmin = np.array([data[data['name'] == x]['Tmin'][0] for x in compounds])
Tmax = np.array([data[data['name'] == x]['Tmax'][0] for x in compounds])

# we have an equimolar mixture
x = np.array([0.2, 0.2, 0.2, 0.2, 0.2])

# Given a T, we can compute the pressure of each species like this:

T = 67 # degC
P = 10**(A - B / (T + C))
print P
print np.dot(x, P)  # total mole-fraction weighted pressure

Tguess = 67
Ptotal = 760

def func(T):
P = 10**(A - B / (T + C))
return Ptotal - np.dot(x, P)

Tbubble, = fsolve(func, Tguess)

print 'The bubble point is {0:1.2f} degC'.format(Tbubble)

# double check answer is in a valid T range
if np.any(Tbubble < Tmin) or np.any(Tbubble > Tmax):
print 'T_bubble is out of range!'

# print gas phase composition
y = x * 10**(A - B / (Tbubble + C))/Ptotal

for cmpd, yi in zip(compounds, y):
print 'y_{0:<10s} = {1:1.3f}'.format(cmpd, yi)

[  498.4320267    182.16010994   898.31061294  1081.48181768   837.88860027]
699.654633507
The bubble point is 69.46 degC
y_benzene    = 0.142
y_toluene    = 0.053
y_chloroform = 0.255
y_acetone    = 0.308
y_methanol   = 0.242