The Rachford-Rice equation: application to ternary diagram

Éderson D’Martin Costa

Laboratório Cristália, Desenvolvimento Analítico, Itapira, SP, Brasil.

João Pedro Braga ^** Correspondence email address: jpbraga@ufmg.br

Universidade Federal de Minas Gerais, Departamento de Química, Belo Horizonte, MG, Brasil.

Tie line	z₁	z₂	z₃	x₁(exp)	x₂(exp)	x₃(exp)	x₁(cal)	x₂(cal)	x₃(cal)
1	0.6140	0.0120	0.3740	0.2520	0.0150	0.7330	0.2511	0.0177	0.7312
				0.9760	0.0090	0.0150	0.9814	0.0063	0.0123
2	0.6310	0.0315	0.3375	0.2980	0.0440	0.6580	0.2958	0.0463	0.6579
				0.9640	0.0190	0.0170	0.9677	0.0167	0.0156
3	0.6695	0.0660	0.2645	0.3990	0.0940	0.5070	0.3967	0.0950	0.5083
				0.9400	0.0380	0.0220	0.9385	0.0374	0.0241
4	0.6910	0.0865	0.2225	0.4650	0.1200	0.4150	0.4713	0.1198	0.4089
				0.9170	0.0530	0.0300	0.9151	0.0526	0.0323
5	0.7110	0.0980	0.1910	0.5230	0.1330	0.3440	0.5251	0.1319	0.3430
				0.8990	0.0630	0.0380	0.8955	0.0643	0.0401
6	0.7195	0.1025	0.1780	0.5550	0.1350	0.3100	0.5495	0.1358	0.3147
				0.8840	0.0700	0.0460	0.8856	0.0700	0.0444
7	0.7274	0.1091	0.1635	—	—	—	0.5852	0.1396	0.2753
				—	—	—	0.8695	0.0786	0.0518
8	0.7424	0.1182	0.1394	—	—	—	0.6515	0.1403	0.2081
				—	—	—	0.8333	0.0961	0.0706

\begin{array}{l} \frac{x_{1} d μ_{1} + x_{2} d μ_{2}}{R T} = x_{1} d (\ln a_{1}) + x_{2} d (\ln a_{2}) \\ = x_{1} d (\ln x_{1}) + x_{2} d (\ln x_{2}) + x_{1} d (\ln γ_{1}) \\ + x_{2} d (\ln γ_{2}) \\ = 0 \end{array}

\begin{matrix} γ_{1}^{α} x_{1}^{α} = γ_{1}^{β} x_{1}^{β} \\ γ_{2}^{α} x_{2}^{α} = γ_{2}^{β} x_{2}^{β} \\ γ_{3}^{α} x_{3}^{α} = γ_{3}^{β} x_{3}^{β} \\ x_{1}^{α} P^{α} + x_{1}^{β} P^{β} = z_{1} \\ x_{2}^{α} P^{α} + x_{2}^{β} P^{β} = z_{2} \\ x_{1}^{α} + x_{2}^{α} + x_{3}^{α} = 1 \\ x_{1}^{β} + x_{2}^{β} + x_{3}^{β} = 1 \\ P^{α} + P^{β} = 1 \end{matrix}

\begin{array}{l} \ln γ_{1} = x_{2}^{2} [τ_{21} {(\frac{G_{21}}{x_{1} + x_{2} G_{21}})}^{2} + \frac{G_{12} τ_{12}}{{(x_{2} + x_{1} G_{12})}^{2}}] \\ \ln γ_{2} = x_{1}^{2} [τ_{12} {(\frac{G_{12}}{x_{2} + x_{1} G_{12}})}^{2} + \frac{G_{21} τ_{21}}{{(x_{1} + x_{2} G_{21})}^{2}}] \end{array}

\begin{array}{l} \frac{G^{E}}{x_{1} x_{2} R T} = τ_{21} G_{21} \frac{x_{2} G_{21}}{{(x_{1} + x_{2} G_{21})}^{2}} + x_{2} \frac{G_{21} τ_{12}}{{(x_{2} + x_{1} G_{12})}^{2}} \\ + x_{1} τ_{12} (\frac{G_{12}}{x_{2} + x_{1} G_{12}})^{2} + x_{1} \frac{G_{21} τ_{21}}{{(x_{1} + x_{2} G_{21})}^{2}} \end{array}

\begin{array}{l} \ln γ_{i} = \frac{\sum_{j = 1}^{m} τ_{j i} G_{j i} x_{j}}{\sum_{l = 1}^{m} G_{l i} x_{l}} + \sum_{j = 1}^{m} \frac{x_{j} G_{i j}}{\sum_{l = 1}^{m} G_{l j} x_{l}} \\ \cdot (τ_{i j} - \sum_{i = 1}^{m} x_{i} \frac{\sum_{r = 1}^{m} τ_{r j} G_{r j} x_{r}}{\sum_{l = 1}^{m} G_{l j} x_{l}}) \end{array}

\begin{array}{l} \ln γ_{i} = \frac{τ_{1 i} G_{1 i} x_{1} + τ_{2 i} G_{2 i} x_{2} + τ_{3 i} G_{3 i} x_{3}}{G_{1 i} x_{1} + G_{2 i} x_{2} + G_{3 i} x_{3}} \\ + \frac{x_{1} G_{i 1}}{G_{11} x_{1} + G_{21} x_{2} + G_{31} x_{3}} \\ \cdot (τ_{i 1} \frac{x_{1} τ_{11} G_{11} + x_{2} τ_{21} G_{21} + x_{3} τ_{31} G_{31}}{G_{11} x_{1} + G_{21} x_{2} + G_{31} x_{3}}) \\ + \frac{x_{2} G_{i 2}}{G_{12} x_{1} + G_{22} x_{2} + G_{32} x_{3}} \\ \cdot (τ_{i 2} \frac{x_{1} τ_{12} G_{12} + x_{2} τ_{22} G_{22} + x_{3} τ_{32} G_{32}}{G_{12} x_{1} + G_{22} x_{2} + G_{32} x_{3}}) \\ + \frac{x_{3} G_{i 3}}{G_{13} x_{1} + G_{23} x_{2} + G_{33} x_{3}} \\ \cdot (τ_{i 3} \frac{x_{1} τ_{13} G_{13} + x_{2} τ_{23} G_{23} + x_{3} τ_{33} G_{33}}{G_{13} x_{1} + G_{23} x_{2} + G_{33} x_{3}}) \\ i = 1, 2, 3. \end{array}

\begin{array}{l} \begin{matrix} (a) Initial guess: x_{i}^{α}, x_{i}^{β}, P^{α} \end{matrix} \\ ↓ \\ \begin{matrix} (b) Calculate \ln γ_{i} and K_{i} (or c_{i}) \end{matrix} \\ ↓ \\ \begin{matrix} (c) Solution of \sum_{i = 1}^{m} \frac{z_{i}}{c_{i} + P^{α}} = 0, for P^{α} \end{matrix} \\ ↓ \\ \begin{matrix} (d) New values: x_{i}^{α} = \frac{K_{i} z_{i}}{1 + P^{α} (K_{i} - 1)}, x_{i}^{β} = \frac{z_{i}}{1 + P^{α} (K_{i} - 1)} \end{matrix} \\ ↓ \\ \begin{matrix} (e) Back to (b) until convergence \end{matrix} \end{array}

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