Bubble point calculation

The bubble point for a mixture of compounds is a useful parameter for designing distillation systems.

At a given pressure P, the bubble point is the temperature at which the first bubble of vapor forms when a liquid mixture is heated. If a gaseous mixture satisfies Dalton's law the total pressure is equal to the sum of the partial pressures of the components of the mixture:

Where \( P_i \) is the partial pressure of component i. Additionally, if Raoult's law is satisfied, the partial pressure can be expressed in terms of the liquid composition and the vapor pressure of the pure compound:

As last ingredient we need a model for the vapor pressure as a function of temperature, in this example we will use an exponential:

Example

A mixture of 50% toluene and 50% benzene is kept at a pressure of 760 [mmHg]. Determine the bubble point of the mixture using the following parameter values for the vapor pressure (Where \( T \) is in Celsius degrees and the vapor pressure value has units of [mmHg]).

Compound	A	B
Benzene	-324.01	10.46
Toluene	-364.43	9.95

One key point of this calculation is to note that at bubble point, the first single bubble is a negligible fraction of the total mass of the system, so the total composition \( z_i \) is equal to the liquid composition \( x_i \). We must find then the value of temperature that satisfies:

For this we use a short python code:

from scipy import optimize
import math

def vapor_press(T, A, B):
    return math.exp(A/T + B)

def system_press(T):
    return 760 - (0.5 * vapor_press(T, -324.01, 10.46) + 0.5 * vapor_press(T, -364.43, 9.95))

T_b = optimize.bisect(system_press, 85, 110)
print('Bubble point temperature: {:.2f} C'.format(T_b))

The result is 93.57 [C]. Due to the problem being nonlinear, we used the bisection algorithm to find a solution, it looks for a root in a specified interval. The bubble point must be located between the boiling point of both pure compounds, which can be obtained from:

Solving for each compound separately you obtain the approximate limits 85 [C] for benzene and 110 [C] for toluene.