CO2-Pentane Binary Phase Behavior: 24-Pet-A7 Secondary and Enhanced Oil Recovery, May 2013, Q2

Problem 2: CO2-Pentane Binary Phase Behavior (20 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Check: Figure 1 is an analog chart (Poettmann & Katz-type binary P-T diagram, reproduced in Whitson & Brulé's Phase Behavior). All numeric readings below (bubble/dew-branch pressures, peak/critical-point locations) are read visually off the printed chart and are good to the chart's own hand-drawn precision (roughly ±20-30 psia, ±5°F) — not exact analytic values. Pure CO₂'s known critical point ($T_c=87.9\,{}^\circ\mathrm{F}$, $P_c=1071$ psia) matches the chart's leftmost (100.00 mol% CO₂) loop closely.

Given. One mole of 70 mol% CO₂ / 30 mol% pentane; Figure 1's family of P-T loops for fixed-composition binary mixtures (bubble branch = rising/left side of each loop, dew branch = falling/right side, each loop's own peak = that composition's critical point; the "Critical Locus" curve connects consecutive peaks).

Find. (a) $P_c$ of the vapor-phase composition and $T_c$ of the liquid-phase composition at (230°F, 825 psia); (b) equilibrium liquid/vapor compositions and moles of liquid; (c) phase state at (230°F, 1000 psia); (d) whether (230°F, 1345 psia) is closer to critical, quantitatively.

Approach. At a fixed (T, P) inside the overall mixture's two-phase region, the coexisting liquid and vapor each sit exactly on the bubble branch (liquid) or dew branch (vapor) of some other fixed composition's own P-T loop — find which two loops cross (230°F, 825 psia), read their compositions and critical points, then apply the lever rule for the mole split.

Identify the tie-line compositions. Tracing Figure 1 at $T=230\,{}^\circ\text{F}$, the bubble branch of the 66.87 mol% CO₂ loop passes almost exactly through $P=825$ psia, and the dew branch of the 89.61 mol% CO₂ loop (descending from its own peak just past $T=205$-$210\,{}^\circ\text{F}$) passes through the same point. These bracket the overall composition (66.87 % $\lt$ 70 % $\lt$ 89.61 %), as equilibrium tie lines must: $$x_{\text{liq}}=0.6687\ \text{mol fraction CO}_2\ (33.13\%\text{ pentane}),\qquad y_{\text{vap}}=0.8961\ \text{mol fraction CO}_2\ (10.39\%\text{ pentane}).$$
Critical points of those two curves (Part a). The vapor lies on the 89.61% loop, whose own peak (critical point) reads at approximately $(T,P)\approx(207\,{}^\circ\text{F},\,1450\text{ psia})$; the liquid lies on the 66.87% loop, whose peak reads at approximately $(320\,{}^\circ\text{F},\,1140\text{ psia})$. Hence $$\boxed{P_{c,\text{vapor phase}}\approx1450\ \text{psia}\ (\text{at }T_c\approx207\,{}^\circ\text{F}),\qquad T_{c,\text{liquid phase}}\approx320\,{}^\circ\text{F}\ (\text{at }P_c\approx1140\text{ psia}).}$$
Moles of liquid (Part b, lever rule). With overall $z_{CO_2}=0.70$ and $n_{total}=1$ mol, $z=x\,L+y\,V$ with $L+V=1$ gives $$L=\frac{y-z}{y-x}=\frac{0.8961-0.70}{0.8961-0.6687}=\frac{0.1961}{0.2274}=\boxed{0.862\ \text{mol liquid}\ (86.2\%),\qquad V=0.138\ \text{mol vapor}\ (13.8\%).}$$ (Mostly liquid, consistent with the overall composition sitting much closer to the liquid tie line than to the vapor one.)
State at 1000 psia, 230°F (Part c). The overall 70% CO₂ mixture's own bubble pressure at 230°F is not drawn directly, but interpolating linearly in composition between the 66.87% curve (bubble $=825$ psia at 230°F, established above) and the 79.18% curve (bubble branch approaching its own nearby peak of $\approx1380$ psia at 245°F, so already $\gtrsim1200$-$1380$ psia by 230°F) for the 70% mixture (only 3.13 of the 12.31-percentage-point gap from 66.87% to 79.18%) gives $$P_{bubble,70\%}\approx825+\frac{3.13}{12.31}(1200\text{ to }1380-825)\approx920\text{-}970\ \text{psia}.$$ Since $1000\ \text{psia}$ exceeds this interpolated bubble pressure at every reasonable reading of the 79.18% branch, the state point sits above the mixture's own bubble curve: $$\boxed{\text{single-phase (compressed) liquid, no vapor present.}}$$
Closer to critical at 1345 psia? (Part d). At 230°F, 1345 psia sits just below both the 89.61% loop's peak (1450 psia at 207°F) and the 79.18% loop's peak (1380 psia at 245°F) — close enough to both that the tie line there runs liquid $\approx$79.18% CO₂ (bubble branch, just short of its own 245°F peak) to vapor $\approx$89.61% CO₂ (dew branch, just past its own 207°F peak). The tie-line length (compositional spread between the phases) is therefore $$\Delta y_{1345}=0.8961-0.7918=0.1043\ (10.4\text{ points}),$$ versus $\Delta y_{825}=0.8961-0.6687=0.2274$ (22.7 points) in Part b — roughly half the spread. $$\boxed{\text{Yes: the two phases at 1345 psia are quantitatively closer to critical (}\Delta y\text{ about half of Part b's), consistent with 1345 psia sitting much nearer the critical-locus curve at this temperature.}}$$

Quantity	Value
Liquid composition at (230°F, 825 psia)	66.87 mol% CO₂ / 33.13 mol% C₅
Vapor composition at (230°F, 825 psia)	89.61 mol% CO₂ / 10.39 mol% C₅
$P_c$ of vapor-phase composition	≈ 1450 psia (at $T_c\approx207\,{}^\circ\text{F}$)
$T_c$ of liquid-phase composition	≈ 320°F (at $P_c\approx1140$ psia)
Moles of liquid / vapor (of 1 mol total)	0.862 mol / 0.138 mol
State at (230°F, 1000 psia)	single-phase liquid
Tie-line spread at 825 psia vs. 1345 psia	22.7 pts → 10.4 pts (closer to critical)

Topic: Binary mixture P-T phase diagrams, tie lines, the critical locus, and the lever rule.

Key relations: lever rule $z=xL+yV$, $L+V=1$; each fixed-composition P-T envelope's own peak is that composition's critical point; the critical locus is the envelope of all such peaks as composition varies continuously from one pure component to the other.

Why this works: at equilibrium, liquid and vapor are each individually saturated at the system (T, P) — the liquid is at its own bubble point and the vapor at its own dew point — so each must lie exactly on the phase-envelope curve of a mixture with that phase's own composition, not the overall feed composition. As pressure at fixed T is raised toward the critical locus, the bubble and dew branches of progressively closer compositions converge, shrinking the tie line to zero exactly at the critical point.

Common pitfall: reading the liquid and vapor compositions off the same loop (the overall 70% CO₂ composition, which isn't drawn) instead of the two different loops whose bubble/dew branches actually cross the state point; forgetting that the "critical pressure of the vapor phase" means the critical point of the vapor's own composition curve, not the system pressure (825 psia) itself.

Source: Whitson & Brulé, Phase Behavior, SPE Monograph Vol. 20, Ch. 3 (binary systems, critical loci); Ahmed, Reservoir Engineering Handbook, 5th ed., Ch. 5 (vapor-liquid equilibria).

Question 2 of 4: CO2-Pentane Binary Phase Behavior

Problem 2: CO2-Pentane Binary Phase Behavior (20 marks)