Flash Distillation Graphical Solution

Reference ID: MET-1B7F | Process Engineering Reference Sheets Calculation Guide

Introduction & Context

Flash distillation, also known as equilibrium or single-stage distillation, 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, causing a portion of the liquid to "flash" into vapor.

This calculation is critical for determining the equilibrium compositions of the resulting vapor and liquid streams. It is widely utilized in the petrochemical, food processing, and pharmaceutical industries to design flash drums, separators, and pre‑treatment stages for distillation columns. By solving the intersection of the flash distillation operating line equation and the equilibrium curve, engineers can predict the separation efficiency of a single‑stage flash system.

Methodology & Formulas

The flash distillation model relies on the simultaneous solution of the mass balance (operating line) and the phase equilibrium relationship. The system is defined by the following governing equations:

The mass balance for the more volatile component is expressed as:

\[ F \cdot x_{F} = V \cdot y + B \cdot x \]

Rearranging this into the slope-intercept form of the operating line yields:

\[ y = -\left( \frac{B}{V} \right) \cdot x + \left( \frac{F}{V} \right) \cdot x_{F} \]

The equilibrium relationship for a binary mixture, assuming a constant relative volatility (α), is defined as relative volatility calculation:

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

To find the intersection, we equate the operating line and the equilibrium relationship, resulting in a quadratic equation of the form \( a \cdot x^{2} + b \cdot x + c = 0 \), where the coefficients are derived as follows:

\[ a = \left( -\frac{B}{V} \right) \cdot (\alpha - 1) \] \[ b = \left( -\frac{B}{V} \right) + \left( \left( \frac{F}{V} \right) \cdot x_{F} \cdot (\alpha - 1) \right) - \alpha \] \[ c = \left( \frac{F}{V} \right) \cdot x_{F} \]

The liquid composition (x) is determined using the quadratic formula, selecting the root that satisfies the physical constraints of the system:

\[ x = \frac{-b \pm \sqrt{b^{2} - 4 \cdot a \cdot c}}{2 \cdot a} \]

Parameter	Constraint / Condition
Vapor flow rate, \( V \)	\( 0 < V < F \)
Feed Composition	\( 0 \leq x_{F} \leq 1 \)
Discriminant	\( b^{2} - 4 \cdot a \cdot c \geq 0 \)
Physical Root (liquid composition)	\( 0 \leq x \leq x_{F} \)

To determine the compositions, you must plot the operating line on the equilibrium diagram. Follow these steps:

Plot the feed composition x_F on the x-axis.
Draw a line with a slope of -B/V (where B is the bottoms liquid flow rate and V is the vapor flow rate) passing through the point (x_F, x_F).
Identify the intersection of this operating line with the equilibrium curve.
The x-coordinate of the intersection represents the liquid composition x, and the y-coordinate represents the vapor composition y.

The slope -B/V represents the negative ratio of the molar flow rates of the liquid (bottoms) and vapor phases leaving the flash drum. This value is critical because:

It dictates the steepness of the operating line.
It is derived from the overall mass balance and the component mass balance equations.
Changes in the feed enthalpy or pressure directly affect the B/V ratio, thereby shifting the operating line and the resulting product compositions.

For a single-stage flash distillation, the feed quality (q) does not directly appear in the graphical construction. The operating line is determined solely by the material balance and the ratio of vapor to liquid flow rates leaving the drum. The q-factor concept is applicable to multi-stage distillation columns (McCabe-Thiele method), where it defines the slope of the feed line. In flash distillation, the feed condition (e.g., subcooled, saturated) influences the degree of vaporization (V/F) through an enthalpy balance, but the operating line remains fixed once V and B are known.

Worked Example: Flash Distillation Graphical Solution

Consider a binary mixture of soy milk components being separated in a flash distillation column. The feed contains the more volatile component (e.g., aroma compounds) with a mole fraction of 0.4. The feed flow rate is 100.0 mol/h, and the vapor flow rate is 40.0 mol/h. The equilibrium constant (α) for the mixture is 2.5. We aim to find the bottoms composition (x_B) and the vapor composition (y_B) leaving the flash drum.

Knowns

Feed flow rate, \(F = 100.0\) mol/h
Feed composition, \(x_F = 0.4\)
Vapor flow rate, \(V = 40.0\) mol/h
Equilibrium constant, \(\alpha = 2.5\)

Step-by-Step Calculation

Determine Bottoms Flow Rate
The bottoms flow rate is obtained from the overall material balance: \(B = F - V\).
\(B = 100.0 - 40.0 = 60.0\) mol/h.
Find Operating Line Parameters
The operating line for flash distillation is given by the equation:
\[ y = -\frac{B}{V} x + \frac{F}{V} x_F \]
Therefore, the slope is: \[ \text{Slope} = -\frac{B}{V} = -\frac{60.0}{40.0} = -1.5 \]
The intercept is: \[ \text{Intercept} = \frac{F}{V} x_F = \frac{100.0}{40.0} \cdot 0.4 = 2.5 \cdot 0.4 = 1.0 \]
Set Up Quadratic Equation
The vapor-liquid equilibrium relationship is: \[ y = \frac{\alpha x}{1 + (\alpha - 1) x} \]
At the intersection of the operating line and equilibrium curve, we set: \[ -\frac{B}{V} x + \frac{F}{V} x_F = \frac{\alpha x}{1 + (\alpha - 1) x} \]
Rearranging gives: \[ \left( -\frac{B}{V} \right) (\alpha - 1) x^2 + \left[ -\frac{B}{V} + \frac{F}{V} x_F (\alpha - 1) - \alpha \right] x + \frac{F}{V} x_F = 0 \]
Substituting the known values:
- \(a = (-1.5) \cdot (1.5) = -2.25\)
- \(b = -1.5 + (1.0 \cdot 1.5) - 2.5 = -2.5\)
- \(c = 1.0\)
Thus, the quadratic equation is: \[ -2.25 x^2 - 2.5 x + 1.0 = 0 \]
Compute Discriminant
The discriminant \(\Delta\) is: \[ \Delta = b^2 - 4ac = (-2.5)^2 - 4 \cdot (-2.25) \cdot 1.0 = 6.25 + 9.0 = 15.25 \]
\[ \sqrt{\Delta} \approx 3.905 \]
Solve for Bottoms Composition \(x_B\)
Using the quadratic formula: \[ x = \frac{-b \pm \sqrt{\Delta}}{2a} \]
The two roots are:
- With the positive discriminant: \(x_{B,1} = \frac{-(-2.5) + 3.905}{2 \cdot (-2.25)} = \frac{2.5 + 3.905}{-4.5} = \frac{6.405}{-4.5} \approx -1.423\) (not physically possible since composition must be between 0 and 1)
- With the negative discriminant: \(x_{B,2} = \frac{2.5 - 3.905}{-4.5} = \frac{-1.405}{-4.5} \approx 0.312\) (valid, and \(0 \leq x_{B} \leq x_F\))
Therefore, the bottoms composition is: \[ x_B = 0.312 \]
Determine Vapor Composition \(y_B\)
Using the equilibrium relationship: \[ y_B = \frac{\alpha x_B}{1 + (\alpha - 1) x_B} = \frac{2.5 \cdot 0.312}{1 + (1.5) \cdot 0.312} = \frac{0.780}{1 + 0.468} = \frac{0.780}{1.468} \approx 0.531 \]
Rounded to three decimal places: \(y_B = 0.531\).

Final Answer

After the flash distillation, the bottoms stream composition is \(x_B = 0.312\) (mole fraction of the more volatile component), and the vapor stream composition is \(y_B = 0.531\) (mole fraction).

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