Flash Distillation Energy Balance

Reference ID: MET-AFBE | Process Engineering Reference Sheets Calculation Guide

Introduction & Context

Flash distillation, or equilibrium flash vaporization, is a fundamental unit operation in process engineering used to separate a multi-component liquid mixture into vapor and liquid phases. The process involves heating a pressurized feed stream and subsequently reducing its pressure across a valve or orifice into a flash drum. This pressure drop causes a portion of the liquid to vaporize instantaneously (flash).

The energy balance for this system is critical for determining the required preheat temperature of the feed to achieve a specific vapor fraction. It is widely used in refinery operations, natural gas processing, and chemical manufacturing to design separation trains and optimize energy efficiency in heat exchanger networks.

Methodology & Formulas

The calculation follows a sequential approach, coupling flash distillation material balance with thermodynamic equilibrium and energy conservation principles.

1. Material Balance and Equilibrium

For a binary system with feed composition \(z\), vapor fraction \(\psi\), and liquid composition \(x\), the operating line is defined by the mass balance. The liquid composition is determined by the intersection of the operating line and the equilibrium curve:

\[ y = \frac{\alpha \cdot x}{1 + (\alpha - 1) \cdot x} \] \[ y = -\left(\frac{1 - \psi}{\psi}\right) \cdot x + \frac{z}{\psi} \]

Solving for \(x\) results in a quadratic equation of the form \(a \cdot x^2 + b \cdot x + c = 0\), where the coefficients are derived from the relative volatility \(\alpha\), feed composition \(z\), and vapor fraction \(\psi\).

2. Flash Temperature Determination

The flash temperature \(T_{flash}\) is the temperature at which the mixture exists at the drum pressure \(P\). This is solved iteratively using Raoult's Law:

\[ P = x \cdot P_{A}^{sat}(T_{flash}) + (1 - x) \cdot P_{B}^{sat}(T_{flash}) \]

Where \(P_{i}^{sat}\) is calculated using the Antoine equation: \(\log_{10}(P_{i}^{sat}) = A_i - \frac{B_i}{T_{flash} + C_i}\).

3. Energy Balance

Under adiabatic conditions, the enthalpy of the feed must equal the sum of the enthalpies of the product streams. The mixture enthalpies are calculated based on the saturated liquid and vapor enthalpies of the pure components:

\[ h_L = x \cdot h_{A,\ell}^{sat} + (1 - x) \cdot h_{B,\ell}^{sat} \] \[ h_V = y \cdot h_{A,v}^{sat} + (1 - y) \cdot h_{B,v}^{sat} \] \[ h_F = \psi \cdot h_V + (1 - \psi) \cdot h_L \]

The feed preheat temperature \(T_F\) is then determined by equating the feed enthalpy to the required mixture enthalpy, assuming the feed remains a subcooled liquid:

\[ T_F = \frac{h_F}{z \cdot c_{p,\ell,A} + (1 - z) \cdot c_{p,\ell,B}} \]

Empirical Range and Validity

Parameter	Constraint/Condition
Pressure	\(P < 10 \text{ bar}\) (Raoult's Law validity)
Vapor Fraction	\(0 < \psi < 1\)
Thermodynamics	Ideal liquid and vapor phases assumed
Heat Capacity	Constant \(c_p\) valid for \(\Delta T < 150^\circ\text{C}\)

To determine the feed enthalpy, you must account for the state of the feed stream relative to the flash drum pressure and temperature. Follow these steps:

Calculate the enthalpy of the liquid phase at the feed temperature and pressure.
If the feed is a subcooled liquid, include the sensible heat required to reach the bubble point.
If the feed is a two-phase mixture, calculate the weighted average enthalpy based on the vapor fraction.
Ensure all reference states for enthalpy calculations are consistent across both the feed and product streams.

The energy balance is essential for solving the flash problem because the system temperature is typically unknown. It serves to:

Close the system of equations by relating the vapor fraction to the heat duty.
Determine the final equilibrium temperature of the flash drum.
Validate that the heat added or removed from the feed matches the enthalpy change of the resulting vapor and liquid phases.

The heat duty term does not appear directly in the Rachford-Rice equation, but it dictates the temperature at which the equation is solved. The process engineer must:

Iterate on the temperature \(T\) until the energy balance is satisfied.
Use the calculated temperature to determine the equilibrium K-values.
Update the vapor fraction based on the new K-values until the Rachford-Rice objective function converges to zero.

Worked Example: Flash Distillation Energy Balance for Benzene–Toluene

A flash drum operating at \(P_{\text{drum}} = 1.013\ \text{bar}\) separates a binary mixture of benzene (A) and toluene (B). The feed is a subcooled liquid at high pressure with composition \(z = 0.5\) (mole fraction benzene). The target vapor fraction is \(\psi = 0.5\) (mole basis). Use constant liquid heat capacities (\(c_{p,\ell,A} = 145.0\ \text{J/(mol·K)}\), \(c_{p,\ell,B} = 165.0\ \text{J/(mol·K)}\)), latent heats (\(\lambda_A = 31000.0\ \text{J/mol}\), \(\lambda_B = 33000.0\ \text{J/mol}\)), and relative volatility \(\alpha = 2.5\). All enthalpies are referenced to saturated liquid at 0°C.

Knowns

Feed composition: \(z = 0.5\)
Drum pressure: \(P_{\text{drum}} = 1.013\ \text{bar}\)
Target vapor fraction: \(\psi = 0.5\)
Relative volatility: \(\alpha = 2.5\)
Liquid heat capacities: \(c_{p,\ell,A} = 145.0\ \text{J/(mol·K)}\), \(c_{p,\ell,B} = 165.0\ \text{J/(mol·K)}\)
Latent heats: \(\lambda_A = 31000.0\ \text{J/mol}\), \(\lambda_B = 33000.0\ \text{J/mol}\)

Step-by-Step Calculation

Determine outlet compositions \(x\) and \(y\). The operating line for \(\psi = 0.5\) and \(z = 0.5\) is \(y = -x + 1\). Intersect with the equilibrium curve \(y = \dfrac{\alpha x}{1 + (\alpha - 1)x}\). Solving the resulting quadratic (with \(\alpha = 2.5\)) yields the liquid composition \(x = 0.387\) and, from equilibrium, the vapor composition \(y = 0.613\).
Find flash temperature \(T_{\text{flash}}\). Use Raoult’s law: \(P_{\text{drum}} = x P_A^{\text{sat}}(T) + (1 - x) P_B^{\text{sat}}(T)\). The Antoine equation (constants from NIST) is solved by bisection between 50°C and 150°C. The computed flash temperature is \(T_{\text{flash}} = 96.89\,^{\circ}\text{C}\).
Compute phase enthalpies at \(T_{\text{flash}}\).
Saturated liquid enthalpies: \(h_{A,\ell}^{(sat)} = c_{p,\ell,A} \cdot T_{\text{flash}} = 14049.05\ \text{J/mol}\), \(h_{B,\ell}^{(sat)} = c_{p,\ell,B} \cdot T_{\text{flash}} = 15986.85\ \text{J/mol}\).
Saturated vapor enthalpies: \(h_{A,v}^{(sat)} = h_{A,\ell}^{(sat)} + \lambda_A = 45049.05\ \text{J/mol}\), \(h_{B,v}^{(sat)} = h_{B,\ell}^{(sat)} + \lambda_B = 48986.85\ \text{J/mol}\).
Mixture enthalpies: \(h_L = x \cdot h_{A,\ell}^{(sat)} + (1 - x) \cdot h_{B,\ell}^{(sat)} = 15236.096\ \text{J/mol}\), \(h_V = y \cdot h_{A,v}^{(sat)} + (1 - y) \cdot h_{B,v}^{(sat)} = 46574.656\ \text{J/mol}\).
Required feed enthalpy from the energy balance. For steady-state adiabatic operation: \(h_F = \psi h_V + (1 - \psi) h_L = 0.5 \cdot 46574.656 + 0.5 \cdot 15236.096 = 30905.376\ \text{J/mol}\).
Solve for feed preheat temperature \(T_F\). The feed is a liquid of composition \(z\); its enthalpy is \(h_F(T_F) = \big[ z \cdot c_{p,\ell,A} + (1 - z) \cdot c_{p,\ell,B} \big] \cdot T_F = c_{p,\text{mix}} \cdot T_F\). With \(c_{p,\text{mix}} = 155.0\ \text{J/(mol·K)}\) and \(h_F = 30905.376\ \text{J/mol}\), \(T_F = 30905.376 / 155.0 = 199.389\,^{\circ}\text{C}\).

Final Answer

The feed must be preheated to approximately \(T_F = 199.389\,^{\circ}\text{C}\) (as a subcooled liquid at sufficiently high pressure) to achieve a vapor fraction \(\psi = 0.5\) in the flash drum at 1.013 bar.

"Un projet n'est jamais trop grand s'il est bien conçu." — André Citroën

"La difficulté attire l'homme de caractère, car c'est en l'étreignant qu'il se réalise." — Charles de Gaulle