Prediction of Saturation Pressures from Serrin’s Equation of State Using the Generalized Maxwell’s Rule

Abstract

A four-parameter equation of state proposed by Serrin is investigated for its ability to predict accurately the saturation pressures of pure substances. The area rule or the generalized Maxwell’s rule is used for the calculation. An algorithm has been proposed by Serrin for implementing the calculations. The algorithm as it appears in print is found to yield results which are vastly different from those presented in Serrin’s article. As a result, a corrected algorithm has been derived and presented in the present article. Although the structure of the algorithm remains intact, the various coefficients and equations in the algorithm are found to differ from what appears in print. Saturation pressure predictions of the corrected algorithm are found to match very closely the predictions of an independent MATLAB-based algorithm developed in this work thereby providing validation of the corrections to Serrin’s algorithm proposed in this article. The corrected algorithm predicts saturation pressures that are close to the experimental values (error within 10%) for water when the parameters of the equation state as predicted by the critical point criteria are used. The sensitivity of the predictions to the specific heat model is also discussed.

Appendix: Algorithm RD

The following steps are adopted in the algorithm:

• 1. For a given dimensionless temperature ($\theta _{1}$), the range within which the dimensionless saturation pressure is likely to exist is found by determining $\pi _{\min ,1}$ and $\pi _{\max ,1}$, the (dimensionless) pressures at which the $\pi \operatorname{versus} \nu$ curve attains a minimum and maximum. These values are found by setting $\partial \pi /\partial \nu = 0$ where $\pi$ is given by Eq. (15). The resulting equation in $\nu$ is solved by using the “roots” function in MATLAB. The appropriate roots are then used to find $\pi _{\min ,1}$ and $\pi _{\max,1}$.

• 2. The dimensionless saturation pressure $\pi _{1}^{\operatorname{sat}}$ is then found by using the “fzero” function in MATLAB. For this purpose, a function termed “Psat_Solution” is created whose algorithm is as follows:

  • i) For a given dimensionless temperature ($\theta _{1}$), start with an initial guess for $\pi _{1}^{\operatorname{sat}}$ in the range($\pi _{\min ,1}$, $\pi _{\max ,1}$). Calculate the corresponding three solutions for dimensionless specific volume at this pressure using the “roots” function.

  • ii) The smallest volume in i) taken to be to $\nu _{1}^{L}$ and the largest to be $\nu _{1}^{V}$.

  • iii) Calculate “Total”, the left-hand side of Eq. (39) and return the value to the function “fzero”. The function iteratively solves for $\pi _{1}^{\mathrm{sat}}$ until the value of the quantity “Total” becomes 0 to within a tolerance limit which was set as $1 \times 10^{ - 6}$. The values of the corresponding dimensionless liquid and vapor specific volumes are the final values of $\nu _{1}^{L}$ and $\nu _{1}^{V}$.

  • iv) The actual physical variables are calculated from Eq. (36).

Note: Results in Tables 3, 7 and 8 corresponding to algorithm RD were obtained using the above procedure. Results in Tables 3 and 7 were also tested against an alternate MATLAB algorithm using the function “fsolve”. This involved solving, at a given value of $\theta _{1}$, the following three equations simultaneously with suitable initial guesses:

• 1. Eq. (15) with $\pi = \pi _{1}^{\operatorname{sat}}$ and $\nu = \nu _{1}^{V}$.

• 2. Eq. (15) with $\pi = \pi _{1}^{\operatorname{sat}}$ and $\nu = \nu _{1}^{L}$.

• 3. Eq. (21) [which is equivalent to Eq. (39)].

The values of the saturation pressures obtained with the two methods matched each other at least up to the third decimal place. The authors thank Dr. Madhucchanda Bhattacharya for carrying out the calculations with “fsolve”.