Bubble-point temperature (Raoult + Antoine)

At the bubble point the vapor mole fractions sum to one; with Raoult's law and Antoine vapor pressures this is one nonlinear equation in temperature.

Governing equation

$T$

Bubble-point temperature (K) — the unknown.

$x_i$

Liquid mole fraction of component i (given).

$P$

System pressure (Pa).

$P^{\mathrm{sat}}_i$

Saturation pressure of component i (Pa), Antoine form.

$a_i,\,b_i,\,c_i$

Antoine constants for \ln P^{\mathrm{sat}}[\mathrm{Pa}].

Derivation

1. 1
   $$y_i P = x_i\,P^{\mathrm{sat}}_i(T)$$

   Raoult's law: at low pressure the partial pressure of i in the vapor equals its liquid mole fraction times its saturation pressure.

2. 2
   $$y_i = x_i\,\frac{P^{\mathrm{sat}}_i(T)}{P} = x_i\,K_i(T)$$

   Divide by P to define the temperature-dependent K-value.

3. 3
   $$\sum_i y_i = \sum_i x_i\,K_i(T) = 1$$

   The bubble point is the temperature at which the first bubble forms, so the vapor mole fractions just close to unity.

4. 4
   $$F(T) = \sum_i x_i\,\frac{1}{P}\exp\!\left(a_i - \frac{b_i}{T+c_i}\right) - 1 = 0$$

   Insert the Antoine vapor pressure to get one monotone-increasing equation in T.

Jacobian

The matrix $J_{ij} = \partial F_i / \partial x_j$ of the residual vector with respect to the unknowns, at the representative operating point. Every value below is computed by the exact forward-mode autodiff engine the solver uses — click a non-zero cell to see the symbolic derivative, each one verified against that engine in our test suite. This is the block a Newton step factorizes.

References

• Poling, Prausnitz & O'Connell, The Properties of Gases and Liquids, 5th ed., Ch. 8.
• NIST Chemistry WebBook, Antoine parameters for ethanol and water.