Methods for Plotting the Equilibrium Line in Absorption and Desorption Operations

In chemical separation engineering, absorption and desorption stand as cornerstone unit operations, where efficiency hinges directly on the phase equilibrium relationship between gas and liquid phases. The construction of the equilibrium line—a curve defining the concentration relationship between gas and liquid under specific operating conditions—is the foundational step in designing separation equipment such as packed towers or tray columns. This article explores the universal principles governing these lines, contrasting the logical frameworks used in absorption versus desorption to provide a comprehensive guide for engineering practice.

Core Thermodynamic Foundations and Equilibrium Equations

Regardless of whether the process involves absorption or desorption, the physical essence of plotting the equilibrium line stems from the thermodynamic equilibrium state of the gas and liquid phases at a given temperature and pressure. For dilute solution systems, Henry's Law is the standard mathematical model used to describe this phase equilibrium. This law posits that, at a constant temperature, the partial pressure $p$ of a solute in the gas phase is directly proportional to its mole fraction $x$ in the liquid phase, with the proportionality constant known as the Henry's Law constant ($H$).

The mathematical expression is defined as:
$$ p = H \cdot x $$

When gas-phase concentration is expressed as mole fraction $y$, Dalton's Law of partial pressures ($p = p_{total} \cdot y$) allows the equilibrium relationship to be transformed into:
$$ y = m \cdot x $$

Here, $m = \frac{H}{p_{total}}$ represents the phase equilibrium constant (or slope). This linear relationship forms the basic framework of the equilibrium line. It is crucial to note that the value of $m$ is highly sensitive to operating conditions: it decreases as temperature rises and increases as total pressure rises. These variations dictate the slope of the equilibrium line, serving as the critical parameter distinguishing different process scenarios.

Equilibrium Line Characteristics in Absorption Operations

In absorption, the solute within a gas mixture is transferred into a liquid solvent, moving from the gas phase to the liquid phase. In this context, the equilibrium line represents the correspondence between gas and liquid concentrations when the system reaches thermodynamic equilibrium.

1. Linearity: Under the assumption of dilute solutions with constant temperature and pressure, the equilibrium line manifests as a straight line passing through the origin. Its slope, $m$, reflects the distribution tendency of the solute between the two phases.
2. Distinction from the Operating Line: The actual separation process does not occur entirely at equilibrium; rather, it is driven by a mass transfer driving force. The operating line describes the relationship between actual gas and liquid compositions at any cross-section of the tower. Its slope is determined by the material balance and typically equals the liquid-to-vapor ratio ($L/V$).
3. Significance in Plotting: On a $y-x$ diagram, the equilibrium line lies above the operating line for low-concentration absorption. The vertical distance between these two lines represents the driving force (concentration difference) at that specific cross-section. A larger driving force translates to a faster mass transfer rate.

Equilibrium Line Characteristics in Desorption Operations

Desorption (or stripping) is the inverse of absorption, aiming to release the solute back into the gas phase from a rich liquid solvent. While the thermodynamic relationship describing the equilibrium state of the system remains unchanged, the reversed operational direction results in distinct manifestations in engineering applications.

1. Slope Invariance: For a given system and set of conditions, the equilibrium relationship $y = mx$ itself does not change with the direction of operation. The equilibrium line remains that fixed reference curve.
2. Shift in Operating Line Position: In a stripper, the solvent enters at the top (low concentration $x_2$), while the gas enters at the bottom (high concentration $y_1$). Consequently, the operating line shifts to lie below the equilibrium line.
3. Driving Force Analysis: The driving force in desorption is characterized by the operating line points falling below the equilibrium line, meaning the actual gas concentration is greater than the gas concentration in equilibrium with the liquid ($y_{gas} > y^*_{liquid}$). Plotting the equilibrium line here is primarily used to determine the number of theoretical stages or packing height, ensuring sufficient driving force to "strip" the solute from the liquid phase.

Comparative Analysis and Application Overview

Comparing absorption and desorption reveals fundamental differences in how equilibrium lines are applied, primarily concerning the relative position of the operating line and the design objectives.

• Relative Position Contrast:
  • Absorption: The operating line is situated above the equilibrium line. The design goal is to keep the operating line as close to the equilibrium line as possible (to maximize separation precision) while strictly maintaining a positive driving force.
  • Desorption: The operating line is situated below the equilibrium line. The objective remains maintaining the minimum driving force, but this is constrained by the energy input required for solvent regeneration.
• Sensitivity to Slope:
  • In absorption, if the liquid-to-vapor ratio ($L/V$) is too low, the operating line may touch the equilibrium line, making it impossible to meet specified separation requirements. In such cases, increasing $L/V$ is necessary.
  • In desorption, if the vapor-to-liquid ratio ($V/L$) is insufficient (indicating inadequate steam usage), the operating line may intersect the equilibrium line, resulting in incomplete desorption. Increasing steam flow or lowering the temperature becomes the corrective action.
• Engineering Application Differences:
  • Absorption towers typically aim for high-purity gas. When the equilibrium line slope $m$ is small (indicating difficult-to-absorb gases), the operating line struggles to approach the equilibrium line, increasing separation difficulty.
  • Desorption towers usually target high-purity solvent regeneration. When the equilibrium line slope $m$ is large (indicating easily desorbed gases), the process is more straightforward and generally requires lower energy consumption.

Conclusion and Practical Plotting Recommendations

In summary, the equilibrium line serves as the vital bridge connecting thermodynamic equilibrium with engineering operation. When plotting this line, the primary task is to accurately determine or look up the Henry's Law constant $H$ to establish the slope $m$. Although the operational direction differs between absorption and desorption, the equilibrium line acts as the unchanging thermodynamic benchmark in both cases.

For practical engineering plotting or simulation, the following steps are recommended:

1. Define Conditions: Determine the system temperature and pressure to calculate the phase equilibrium constant $m$.
2. Draw the Reference: Plot the straight line $y=mx$ on the $y-x$ coordinate system as the equilibrium line.
3. Plot the Operating Line: Construct the corresponding operating line based on material balance calculations.
4. Verify Driving Force: Check the relative positions of the operating and equilibrium lines to confirm that sufficient mass transfer driving force exists.

Mastering these universal principles enables engineers to rapidly assess separation feasibility under various conditions and provides a solid theoretical basis for subsequent equipment selection and optimization.