Vapor Liquid Equilibrium#

Introduction#

Vapor-liquid equilibrium (VLE) describes the distribution of a chemical species between the vapor phase and a liquid phase in multicomponent phase equilibrium. Vapor-liquid equilibrium information is helpful in designing distillation processes, a fundamental chemical engineering separation technique; distillation is used to separate or partially separate components in a mixture by boiling (vaporization) followed by condensation. However, other types of separation processes rely on vapor-liquid equilibrium. For example, flash separation occurs when a saturated liquid stream undergoes a reduction in pressure by passing through a throttling valve, which gives a mixture in vapor-liquid equilibrium. Finally, liquefaction is a process that generates a liquid from a gas. Major commercial applications of liquefaction are the liquefaction of air, which allows the separation of constituents, such as oxygen, nitrogen, and noble gases, or the conversion of vapor fuels, such as methane, into a liquid form which is more convenient for transport. In all these applications, vapor-liquid equilibrium plays a critical role.

In this lecture we will:

Ideal Vapor-Liquid Equilibrium model#

The fugacity matching condition given by Definition 22, in combination with phase specific fugacity models, gives us tools to design phase equilibrium processes.

Definition 25 (General Vapor-Liquid Equilibrium)

In vapor-liquid equilibrium, let’s consider a system with the composition set \(\mathcal{M}\). The fugacity matching condition, along with the phase-specific models, gives the relationship:

(52)#\[\hat{\phi}^{v}_{i}y_{i}P = \hat{\gamma}_{i}x_{i}\phi^{sat}_{i}P^{sat}_{i}\qquad\forall{i}\in\mathcal{M}\]

On the vapor side of Eqn. (52), the term \(\hat{\phi}^{v}_{i}\) denotes the fugacity coefficient for component \(i\), \(y_{i}\) denotes the mole fraction of component \(i\) in the vapor phase, and \(P\) denotes the system pressure.

On the liquid side of Eqn. (52), the term \(\hat{\gamma_{i}}\) denotes the activity coefficient for component \(i\), \(x_{i}\) denotes the mole fraction of component \(i\) in the liquid, \(P^{sat}_{i}\) denotes the saturation pressure for component \(i\), and \(\phi^{sat}_{i}\) denotes the vapor phase fugacity coefficient for pure component \(i\) at the saturation pressure.

The fugacity coefficient \(\hat{\phi}_{i}^{v}\) and the activity coefficient \(\hat{\gamma}_{i}\) describe the vapor and liquid phase non-ideality. There are a variety of models that can be used to capture the molecular interactions that are responsible for non-ideality. However, for now, we make an essential simplifying assumption typically not valid for a real system; namely, both the vapor and liquid are ideal.

Observation 19 (Ideal vapor and liquid phases)

For ideal vapor or liquid phases, the fugacity and activity coefficients go to one; the Gibbs excess (resdiudal) energy is zero. Thus, Eqn. (52) for ideal phases is given by:

(53)#\[y_{i}P = x_{i}P^{sat}_{i}\qquad\forall{i}\in\mathcal{M}\]

where \(x_{i}\) is a liquid-phase mole fraction, \(y_{i}\) is a vapor-phase mole fraction, and \(P_{i}^{sat}\) is the saturation pressure of pure species \(i\) at the system temperature.

Ideal binary phase diagrams#

A phase diagram is a diagram that shows the conditions at which thermodynamically distinct phases (solid, liquid, or vapor states) occur and coexist at equilibrium. Equation (53) is useful in calculating binary phase diagrams, i.e., the phase diagram for a mixture of two components.

Binary mixtures are convenient because they can easily be visualized. We consider two types of binary phase diagrams: Pressure composition diagram (Pxy) and Temperature composition diagram (Txy).

Pressure composition diagram (Pxy)#

A binary pressure composition diagram, typically called a Pxy diagram, shows the relationship between the phases co-existing in equilibrium for different binary mixtures of components A and B at a fixed temperature and varying pressures. Pressure composition diagrams have two lines, a liquid, and a vapor line.

Ideal liquid line#

The liquid line governs the composition of a mixture of the component set \(\mathcal{M}\) in the liquid phase during vapor-liquid equilibrium.

Observation 20 (Derivation of the liquid line Pxy)

Suppose we have a system composed of the species set \(\mathcal{M}\) with \(\dim\mathcal{M} = M\), in vapor-liquid equilibrium. Further, suppose both the vapor and liquid phases are ideal.

To derive the liquid equilibrium line, we sum the equilibrium matching conditions, given by Eqn (53), over the species set \(\mathcal{M}\):

\[\begin{split} \begin{eqnarray} y_{1}P &=& x_{1}P^{sat}_{1}\\ y_{2}P &=& x_{2}P^{sat}_{2}\\ & \vdots & \\ y_{M}P &=& x_{M}P^{sat}_{M} \end{eqnarray} \end{split}\]

which gives:

\[\left(y_{1}+y_{2}+\dots+y_{M}\right)P = x_{1}P^{sat}_{1}+x_{2}P^{sat}_{2}+\dots+x_{M}P^{sat}_{M}\]

However, we know that \(\left(y_{1}+y_{2}+\dots+y_{M}\right)=1\), thus the liquid equilibrium line is given by:

(54)#\[P = \sum_{i\in\mathcal{M}}x_{i}P^{sat}_{i}\]

For a binary system \(\dim\mathcal{M}=2\) the liquid line given by Eqn (54) is a sum over two components:

(55)#\[P = x_{1}P_{1}^{sat} + (1-x_{1})P_{2}^{sat}\]

where we take advantage of the summation property of mole fraction, i.e., \(\sum_{i\in\mathcal{M}}x_{i} = 1\). The ideal binary liquid line described by Eqn. (55) is line connecting the saturation pressures \(P^{sat}_{1}\) and \(P^{sat}_{2}\) in the P-xy plane.

Ideal vapor line#

The vapor line governs the composition of a mixture of the component set \(\mathcal{M}\) in the vapor phase during vapor-liquid equilibrium.

Observation 21 (Derivation of the vapor line)

Suppose we have a system composed of the species set \(\mathcal{M}\) with \(\dim\mathcal{M} = M\), in vapor-liquid equilibrium. Further, suppose both the vapor and liquid phases are ideal.

To derive the vapor line, which governs the composition of the vapor phase, we solve Eqn. (53) for the mole fraction of component \(i\) in the vapor phase:

\[y_{i} = \frac{x_{i}P^{sat}_{i}}{P}\]

However, we know the pressure \(P\) is governed by Eqn. (55), thus, the vapor phase mole fraction of component \(i\) is given by:

(56)#\[y_{i} = \frac{x_{i}P^{sat}_{i}}{\displaystyle{\sum_{j\in\mathcal{M}}x_{j}P^{sat}_{j}}}\qquad\forall{i\in\mathcal{M}}\]

For a binary system \(\dim\mathcal{M}=2\) the demoninator of the vapor line given by Eqn (56) is a sum over two components:

(57)#\[y_{i} = \frac{x_{i}P^{sat}_{i}}{x_{1}P_{1}^{sat} + (1-x_{1})P_{2}^{sat}}\qquad{i=1,2}\]

The ideal binary vapor line described by Eqn. (57) is a concave upward curve that connects the saturation pressures \(P^{sat}_{1}\) and \(P^{sat}_{2}\) in the P-xy plane.

Equilibrium Tie-line on Pxy diagram#

Pressure composition diagrams (Pxy) are useful because they map regions of liquid, vapor or liquid-vapor equilibrium. For example, consider the binary Acetone(1) - Water(2) system at P \(\simeq\) 130 kPa and T = 80\(^{\circ}\)C (Fig. 19).

../_images/Fig-Pxy-Diagram-Labels.pdf

Fig. 19 Ideal pressure composition (Pxy) diagram for a binary mixture of Acetone(1) and Water(2) at T = 80\(^{\circ}\)C. Antoine parameters were taken from Table B.2 of Smith, Van Ness, Abbott, and Swihart 8th edition. Nomenclature: \(x_{1}^{eq}\) denotes the equilibrium composition in the liquid; \(y_{1}^{eq}\) denotes the equilibrium composition in the vapor.#

We can draw a horizontal line of constant pressure, e.g., at P \(\simeq\) 130 kPa, and estimate the compositions of Acetone(1)-Water(2) that exist as a liquid, a vapor or a mixed system in vapor-liquid equilibrium. The intersection of the pressure line with the liquid line gives the equilibrium composition of the liquid, denoted as \(x^{eq}_{1}\). Likewise, the intersection of the constant pressure line with the vapor line gives the equilibrium composition in the vapor, denoted as \(y^{eq}_{1}\).

The constant pressure line that connects the equilibrium liquid and vapor compositions is called the constant pressure equilibrium tie-line.

Temperature composition diagram (Txy)#

A binary temperature composition diagram, typically called a Txy diagram, shows the relationship between the phases co-existing in equilibrium for different binary mixtures of components A and B at a fixed pressure and varying temperature.

Ideal liquid line#

The liquid line in a Txy diagram has the same functional form as Eqn. (54) however, what is known versus unknown changes. In the case of a Txy diagram, the pressure and composition are known, but the temperature is unknown. Thus, we use the liquid model in Eqn. (54) to solve for the temperature at different pressure composition pairs; the liquid line is a function of temperature because the saturation pressures \(P_{i}^{sat}\) for each component are functions of temperature.

Observation 22 (Estimation of temperature Txy diagram)

Suppose we have a system composed of the species set \(\mathcal{M}\) with \(\dim\mathcal{M} = M\), in vapor-liquid equilibrium. Further, suppose both the vapor and liquid phases are ideal.

To estimate the temperature-composition relationship for the liquid phase components, when given the pressure, we minimize the squared difference between the pressure estimated by Eqn (54) which we denote as \(\hat{P}\) and the known system pressure:

\[\min_{T} \left(P-\hat{P}\right)^{2}\]

subject to bounds on the temperature \(0\leq{T}\leq\infty\). We solve this optimization problem for each liquid phase composition, i.e., for each value of \(x_{i}\).

Ideal vapor line#

The hard work when constructing a Txy diagram is done when computing the liquid line, i.e., repeadtly solving the optimization problem shown in Observation 22. Once this is done, we can easily solve (at least in the ideal case) for the vapor phase composition.

Observation 23 (Derivation of the vapor line Txy)

Suppose we have a system composed of the species set \(\mathcal{M}\) with \(\dim\mathcal{M} = M\), in vapor-liquid equilibrium. Further, suppose both the vapor and liquid phases are ideal, and we have computed the liquid line.

Then, the vapor phase composition can be estimate for components \(i=1,2,\dots,M\) from:

\[y_{i} = \frac{x_{i}P^{sat}_{i}}{P}\qquad\forall{i}\in\mathcal{M}\]

where the system pressure \(P\) is known, and the \(\left(x_{i},T_{i}\right)\) data from liquid line is used to compute the numerator.

Equilibrium Tie-line on Txy diagram#

Temperature composition diagrams (Pxy) are also useful because they map regions of liquid, vapor or liquid-vapor equilibrium at a fixed pressure for different temperatures. For example, consider the binary Acetone(1) - Water(2) system at P = 125 kPa and T \(\simeq\) 83\(^{\circ}\)C (Fig. 21).

../_images/Fig-Txy-Diagram-Labels.pdf

Fig. 21 Ideal temperature composition (Txy) diagram for a binary mixture of Acetone(1) and Water(2) at P = 125 kPa. Antoine parameters were taken from Table B.2 of Smith, Van Ness, Abbott, and Swihart 8th edition. Nomenclature: \(x_{1}^{eq}\) denotes the equilibrium composition in the liquid; \(y_{1}^{eq}\) denotes the equilibrium composition in the vapor.#

We can draw a horizontal line of constant temperature, e.g., at T \(\simeq\) 83\(^{\circ}C\), and estimate the compositions of Acetone(1)-Water(2) that exist as a liquid, a vapor or a mixed system in vapor-liquid equilibrium. The intersection of the temperature line with the liquid line gives the equilibrium composition of the liquid, denoted as \(x^{eq}_{1}\). Likewise, the intersection of the constant temperature line with the vapor line gives the equilibrium composition in the vapor, denoted as \(y^{eq}_{1}\).

The constant temperature line that connects the equilibrium liquid and vapor compositions is called the constant temperature equilibrium tie-line.

Flash separation processes#

In a Flash separation process, a multicomponent saturated liquid composed of the species set \(\mathcal{M}\) enters the Flash drum (at some temperature and pressure). Inside the drum the pressure is rapidly decreased leading to flash vaporization, and a liquid and vapor streams exit the drum (Fig. 22).

../_images/Fig-Flash-Sep-Schematic.pdf

Fig. 22 Schematic of a Flash separation process. Nomenclature: \(\dot{F}\) total mole flow into the unit (units: mol/time); \(\dot{L}\) total mole flow rate of the liquid stream exiting the unit (units: mol/time); \(\dot{V}\) total mole flow rate of the vapor stream exiting the unit (units: mol/time).#

Remark 5 (Flash equilibrium assumption)

The key assumptions we make when modeling a Flash separation process are:

  • The flash drum is well-mixed

  • The contents of the drum are in vapor-liquid equilibrium, and hence, the vapor and liquid streams that exit the drum are in vapor-liquid equilibrium

  • Equilibrium is instantaneous inside the drum

Total and Species mole balance Flash drum#

There are no chemical reactions occuring in the Flash drum, thus, the number of moles is conserved. The total mole balance around the Flash drum is given by:

(58)#\[\dot{F} = \dot{L} + \dot{V}\]

where \(\dot{F}\) denotes the total mole flow into the unit (units: mol/time), \(\dot{L}\) denotes the total mole flow rate of the liquid stream exiting the unit (units: mol/time), and \(\dot{V}\) represents the total mole flow rate of the vapor stream leaving the unit (units: mol/time).

Similarly, we can write a species mole balance around all the species in the species set \(\mathcal{M}\):

(59)#\[\dot{F}z_{i} = \dot{L}x_{i} + \dot{V}y_{i}\qquad\forall{i}\in\mathcal{M}\]

where \(z_{i}\) denotes the mole fraction of component \(i\) in the input stream (we use \(z\) to avoid confusion with the liquid stream exiting the unit), \(x_{i}\) denotes the mole fraction of component \(i\) in the liquid stream exiting the unit, and \(y_{i}\) represents the mole fraction of component \(i\) leaving the unit in the vapor stream.

Let’s develop a strategy that uses these balances to model a Flash separation process (Observation 24).

Algebraic solution strategy#

Observation 24 (Algebraic strategy: Solution of an ideal Flash separation problem)

A liquid feed stream that contains 60% component A (1) and 40% component B (2) is separated in a Flash Drum operating at P = 1.21 MPa and T = 150\(^{\circ}\)C. The saturation pressures of pure components A and B can be modeled using the Antoine equation:

\[ \ln\left(P_{i}^{sat}\right) = A_{i} -\frac{B_{i}}{T+C_{i}}\qquad{i=1,2}\]

where \(T\) denotes the temperature in units \(^{\circ}\)C and \(P_{i}^{sat}\) denotes the saturation pressure in units of kPa for component \(i\), and \(A_{i},B_{i}\) and \(C_{i}\) denote the Antoine parameters for compound \(i\). The assumptions in Remark 5 are applicable to this problem.

Develop a strategy to calculate:

  • The fraction of the input stream that exits the drum as liquid

  • The fraction of the input stream that exits the drum as vapor

  • The composition of the liquid (\(x_{1}\) and \(x_{2}\)) and vapor (\(y_{1}\) and \(y_{2}\)) in the exit streams

Solve for liquid and vapor composition: Let’s start by writing down the governing equations. Starting from the total mole balance:

\[\dot{F} = \dot{L}+\dot{V}\]

we divide both sides by \(\dot{F}\) which gives a relationship for the fraction(s) of liquid and vapor in the outlet streams:

\[\hat{L}+\hat{V} = 1\]

Further, we know that the species mol balance around species \(i\) is given by:

\[\dot{F}z_{i} = \dot{L}x_{i}+\dot{V}y_{i}\qquad{i=1,2}\]

However, because we assumed ideal vapor and liquid, we know that the fugacity matching condition relates the liquid and vapor phase equilibrium compositions:

\[y_{i}P = x_{i}P_{i}^{sat}\qquad{i=1,2}\]

Lastly, we can sum the matching conditions to arrive at an expression that governs the pressure that is only in \(x_{i}\):

\[P = x_{1}P_{1}^{sat} + x_{2}P_{2}^{sat}\]

which can be re-written (because we are in a binary mixture) as:

\[P = x_{1}P_{1}^{sat} + \left(1-x_{1}\right)P_{2}^{sat}\]

Finally, we can re-arrange the pressure expression to solve for \(x_{1}\):

\[x_{1} = \frac{P - P_{2}^{sat}}{P_{1}^{sat} - P_{2}^{sat}}\]

Once we have the liquid phase composition, we can use the fugacity matching condition to relate the liquid phase and the vapor phase because we have assumed they are in equilibrium, i.e.,

\[y_{i} = x_{i}\left(\frac{P_{i}^{sat}}{P}\right)\qquad{i=1,2}\]

Solve for the fraction of liquid and vapor \(\hat{L}\) and \(\hat{V}\) exiting the unit: Now that we have the composition of the exit streams, we can use the total and species mole balances to estimate the fraction of the input feed that exits the unit as liquid \(\hat{L}\) and vapor \(\hat{V}\). Dividing both sides of the species mole balance by \(\dot{F}\), we get the relationship:

\[z_{i} = \hat{L}x_{i}+\hat{V}y_{i}\qquad{i=1,2}\]

which we can use (along with the total mole balances) to solve for \(\hat{L}\) and \(\hat{V}\), i.e.,

\[\begin{split}\begin{eqnarray} \hat{L}x_{1}+\hat{V}y_{1} &=& z_{1} \\ \hat{L}+\hat{V} &=& 1 \end{eqnarray}\end{split}\]

Approach 1: Substitution/elimination: Let’s rearrange the total mol expression to get \(\hat{L}\) in terms of \(\hat{V}\):

\[\hat{L} = 1 - \hat{V}\]

which can be substituted into the species mol balance (let’s use component 1):

\[z_{1} = \left(1 - \hat{V}\right)x_{1}+\hat{V}y_{1}\]

and solved for \(\hat{V}\):

\[\hat{V} = \frac{z_{1} - x_{1}}{y_{1} - x_{1}}\]

Once we have \(\hat{V}\), we can compute \(\hat{L}\) from the total mol balance.

Approach 2: Linear algebra: We can rewrite the 2\(\times\)2 system of equations into matrix-vector form and then compute the matrix inverse. In particular, the species and total mass balances for component 1 can be written as:

\[\begin{split}\begin{pmatrix} x_{1} & y_{1} \\ 1 & 1 \end{pmatrix} \begin{pmatrix} \hat{L} \\ \hat{V} \end{pmatrix} = \begin{pmatrix} z_{1} \\ 1 \end{pmatrix}\end{split}\]

which is the form \(\mathbf{A}\mathbf{x} = \mathbf{b}\), where the solution takes the form: \(\mathbf{x} = \mathbf{A}^{-1}\mathbf{b}\).

Graphical solution strategy#

The graphical solution strategy for flash problems relies on using a Pressure composition diagram (Pxy) or a Temperature composition diagram (Txy) to estimate the equilibrium composition; draw a line of constant pressure (temperature) and calculate the equilibrium composition by finding the points of intersection with the liquid and vapor lines. However, once we have the equilibrium composition, we still need to estimate the liquid and vapor flow rates.

We develop another expression called the Flash Lever Rule (FLR) to estimate the liquid and vapor at equilibrium. The steady-state total mole and species mole balances around the flash unit (for a binary mixture) are given by (see Eqn. (58) and Eqn (59)):

\[\begin{split} \begin{eqnarray} \dot{F} &=& \dot{L} + \dot{V} \\ \dot{F}z_{i} &=& \dot{L}x_{i}+\dot{V}y_{i}\qquad{i=1,2} \end{eqnarray} \end{split}\]

where \(\dot{F}\) is the total mole flow rate in the feed stream into the unit, \(\dot{L}\) is the total mole flow rate in the liquid stream coming out of the unit on the bottom and \(\dot{V}\) is the total mole flow rate in the vapor stream coming out of the unit on the top. Finally, \(z_{i},x_{i}\) and \(y_{i}\) is the mole fraction of component \(i\) in the feed, liquid and vapor streams.

Observation 25 (Graphical strategy: Flash lever rule)

To derive the Flash Lever Rule, substiture \(\dot{F}\) from the total mole balance into the species mole balance:

\[\left(\dot{L}+\dot{V}\right)z_{i} = \dot{L}x_{i}+\dot{V}y_{i}\]

and solve for the ratio of the output streams:

(60)#\[\frac{\dot{L}}{\dot{V}} = \frac{y_{i} - z_{i}}{z_{i} - x_{i}}\qquad{i=1,2}\]

Eqn. (60) is the Flash lever rule. Let’s define: \(a_{i}\equiv z_{i} - x_{i}\) and \(b_{i}\equiv y_{i} - z_{i}\), then:

(61)#\[\frac{\dot{L}}{\dot{V}} = \frac{b_{i}}{a_{i}}\qquad{i=1,2}\]

We can estimate \(a_{i}\) and \(b_{i}\) from the pressure-composition or temperature composition diagrams. However, we still need to estimate the liquid and vapor mole flow rates exiting the unit. To do this, we take advantage of the identity:

\[\frac{\dot{L}}{\dot{V}} = \frac{\hat{L}}{\hat{V}}\]

Thus, we Eqn. (61) becomes:

\[\frac{\hat{L}}{\hat{V}} = \frac{b_{i}}{a_{i}}\qquad{i=1,2}\]

which can be rearranged (in combination with the total mole balance) to give:

\[\hat{V} = \left(\frac{b_{i}}{a_{i}} + 1\right)^{-1}\]

Once we have \(\hat{V}\), we can find \(\hat{L}\) from the total mole balance.

Simple Hampson–Linde liquifaction cycle#

The Simple Hampson–Linde cycle is a process for the liquefaction of gases, e.g., for air separation (Fig. 23).

../_images/Fig-PID-Hampson-Linde-Process.pdf

Fig. 23 Schematic of the Hampson-Linde cycle. Circles denote operating points in the cycle, where \(\mathcal{P}_{ij}\) is the path that connects operating point \(i\) and \(j\).#

n this cycle, a gas is isothermally compressed from low pressure to high pressure (\(\mathcal{P}_{12}\)). Then the high-pressure, high-temperature gas passes through a heat exchanger and is cooled by the low-pressure, low-temperature gas returning from the collection tank (\(\mathcal{P}_{23}\)). Next, the gas passes through a constant enthalpy expansion via a Joule-Thomson valve, a type of throttle valve for gases. Following the Joule-Thomson expansion, the stream returns to its initial low pressure (\(\mathcal{P}_{34}\)). However, the exciting aspect of the isoenthalpic expansion is the phase change; the drop in pressure leads to vapor-liquid equilibrium and a temperature decrease. Thus we have both vapor and liquid phases that are cold relative to the input stream. The low-temperature liquid fraction is collected, while the low-pressure, cool vapor is recycled, begin reheated, and then mixed with the input stream (\(\mathcal{P}_{51}\)).

../_images/Fig-TS-Hampson-Linde-Process.pdf

Fig. 24 Schematic of the Hampson-Linde cycle on a Temperature-Entropy (TS) diagram.#

The Simple Hampson–Linde cycle can be visualized on a Temperature-Entropy (TS) diagram (Fig. 24).

Summary#

In this lecture we: