CHEE3005 – Multicomponent Separations Shortcut methods Notes

Multicomponent Separations: Shortcut Methods

Introduction

• Flash separations are driven by Vapour Liquid Equilibrium (VLE).

• Enhanced separations beyond VLE require multi-stage systems.

• This involves using the top product (D) as feed to a second stage or the bottom product (W) as feed to the next stage.

Multicomponent Distillation Process

• Temperature in each stage determines the ratio of D/F (Distillate to Feed).

• Product flow rate decreases progressively with an increasing number of stages.

Counter-Current Processes

• Rectifying Section: Some of the top product is returned to the top stage as reflux.

• Stripping Section

Distillation Column Components

• Feed: Enters the column.

• Enriching (Rectifying) Section: Located above the feed.

• Stripping Section: Located below the feed.

• Reboiler: Provides heat for vaporization at the bottom.

• Condenser: Condenses vapor at the top.

• Reflux Drum: Collects condensed liquid.

• Top Product: Distillate.

• Bottom Product: Waste.

• Reflux: Liquid returned to the top of the column.

Trade-offs in Distillation

• Distillation columns achieve VLE more efficiently than multiple flash vessels.

• Vapor and liquid are in equilibrium.

• Vapor flows from the stage below, and liquid flows to the tray beneath.

Column Design

• For a fixed column arrangement with specified stages and efficiency, mixture separation can be calculated.

• Often, we specify top/bottom product compositions and design a column to achieve this.

• Key design parameters:

  • Number of stages.

  • Reflux Ratio.

  • Reboiler/Condenser duties.

• This requires considering 'key components' in the mixture.

Key Components

• Consider 5 components (A-E) in order of volatility (A is most volatile).

• Suppose the separation is A,B,C,D in the top product and C,D,E in the bottom product.

• Heavy Key (HK): The heaviest component in the top product (D).

• Light Key (LK): The lightest component in the bottom product (C).

• Designing the column based on heavy and light keys allows other components to largely separate themselves.

• Key components have no physical significance and are chosen for design purposes.

• Used with 'Short-cut' methods for column design.

Column Design Parameters

• Key parameters for column design:

  • Plates or packing.

  • Number of stages or packed height.

  • Column Diameter.

  • Reflux Ratio.

• Reflux ratio determines the operating cost of the column.

Reflux Ratio (L/D)

• High R:

  • More liquid (and vapor) in the column.

  • Higher duties on condenser and reboiler.

• Low R:

  • Little contact between vapor and liquid.

  • Poor Separation.

  • More stages required.

• A minimum reflux ratio ($R_{min}$) exists for a given degree of separation.

• An infinite number of stages are required at $R_{min}$.

Optimal Reflux Ratio

• When R increases, N (number of stages) decreases, reducing capital costs but increasing energy requirements and operating costs.

• When R decreases, N increases, raising capital costs but reducing energy requirements.

• Optimum reflux ratio ($R_{optimum}$) balances capital and operating costs.

• Typically, $R/R_{min} = 1.05 - 1.3$.

Reflux Ratio Calculation (Binary System)

• The q-line is drawn on the x-y diagram.

• At $R=0$, the TOL (Tie-line of Operation) at $R_{min}$ is found by the intersection of the q-line and the equilibrium curve.

• $R_{min}$ is found from the gradient or intercept of the TOL.

• Increasing R increases the gradient of the TOL.

Multicomponent Systems

• Calculations are straightforward for binary systems.

• $R_{min}$ depends on the desired top product composition, the state of the feed, and VLE data.

Shortcut Methods

• To design a distillation column, the number of stages and reflux ratio are needed.

• Solutions for multicomponent systems are too complicated for hand calculations.

• Shortcut methods use numerical/graphical techniques with simple mathematical expressions.

• Often based on empirical data and key components.

Assumptions for Shortcut Methods

• K value data should be readily obtainable.

• No reactions occur.

• Temperatures and pressures should be within the normal operating range.

Empirical Methods

• Gilliland or Erbar-Maddox Correlation: Relates the number of stages (n) and reflux ratio (R) to the minimum number of stages ($n<em>m$) and the minimum reflux ratio ($R</em>m$).

  • $n_m$ occurs at total reflux.

  • $R_m$ corresponds to an infinite number of stages.

  • Note: n refers to N-1, where N is the number of stages excluding the reboiler.

Erbar-Maddox Correlation

• Generally more accurate than the Gilliland correlation.

• Both correlations require knowledge of three of the four parameters (n, $n<em>m$, R, $R</em>m$) to calculate the fourth.

Determining Initial Values

1. Minimum number of theoretical stages, $N_m$.

2. Minimum Reflux Ratio, $R_m$.

3. Number of theoretical stages, N.

• $N_m$ is found using the Fenske Equation at total reflux.

• $R_m$ is found using the Underwood Equation.

• N is found using the Erbar-Maddox Correlation.

Fenske Equation

• Also called the 'Fenske-Underwood' equation.

• Allows the minimum number of stages to be calculated based on specified top and bottom product compositions.

• Often specified in terms of heavy key (HK) and light key (LK), not a general component i.

• Equation: $N<em>m = \frac{ln[(\frac{x</em>{LK,d}}{x<em>{HK,d}})(\frac{x</em>{HK,b}}{x<em>{LK,b}})]}{ln(\alpha</em>{avg})}$

  • $N_m$ = minimum number of stages (including reboiler).

  • d denotes the top product.

  • b denotes the bottom product.

  • $\alpha_{avg}$ = average relative volatility of light key to heavy key.

Calculating $N_m$ using the Fenske Equation

• Varying levels of accuracy are possible depending on the assumptions made.

• Accuracy depends on the difference between the $\alpha$ values of the LK and HK.

• Depends on the variation in the $\alpha$ values with temperature and pressure.

Assumptions for Fenske Equation

• If $\alpha$ does not vary significantly with T:

  • Assume that $\alpha<em>{i,av}$ and $\alpha</em>{r,av}$ equal the values that correspond to the feed conditions.

  • This means that $N_m$ can be calculated directly from the Fenske Equation.

• If there is a wide difference between the relative volatilities at the top and bottom of the column: The use of average value in Fenske Equation will underestimate the number of stages.

  • Calculate the minimum number of stages in the rectifying and stripping stages separately.

  • Reduces the variation in $\alpha$ for each set of calculations.

  • This also allows location of the feed point.

Separate Calculations -

$(\frac{x<em>{i,d}}{x</em>{ref,d}})=\alpha<em>{i,F}^{N</em>m}(\frac{x<em>{i,F}}{x</em>{ref,F}})$
$(\frac{x<em>{i,F}}{x</em>{ref,F}})=\alpha<em>{i,F}^{N</em>m}(\frac{x<em>{i,b}}{x</em>{ref,b}})$

• $N<em>m = 1 + n</em>m$

Minimum Reflux Ratio, $R_m$

• Estimate $R<em>m$ and using an appropriate ratio of $R/R</em>m$, obtain the number of stages, n, from Gilliland or Erbar-Maddox correlations.

• Common method: Underwood Equation (assumes Constant Molar Overflow and Constant Relative Volatilities).
  $R<em>m = \sum</em>{i=1}^{n} \frac{\alpha<em>{i,av} x</em>{i,d}}{\alpha_{i,av} - \theta} - 1$

• Where $R<em>m$ is the minimum reflux ratio, $x</em>{i,d}$ is the concentration of component i in the tops at minimum reflux, $\alpha_{i,av}$ is the constant relative volatility at average temperature.

Underwood Equation

• q is the solution to the equation:

$\sum<em>{i=1}^{n} \frac{\alpha</em>{i,av} z<em>i}{\alpha</em>{i,av} - \theta} = 1 - q$

*   q = (Heat to vaporize 1 mole of feed) / (Molar latent heat of feed).
*    $z_i$ = Molar fraction of component i in the feed.

• q depends on the saturation condition of the feed.

  • 1-q is the fraction of feed that is vapor.

  • q represents a relative volatility.

• q must lie between $\alpha<em>{HK}$ and $\alpha</em>{LK}$.

• Iterative calculations are required to establish q and then $R_m$.

• Then use Gilliland or Erbar-Maddox correlations to relate n and R.

Location of Feed Point- Kirkbride Equation

• The Kirkbride Equation yields the ratio of the number of theoretical stages in the rectifying section, m to the number in the stripping section, p.
  Knowing the ratio m/p and the total : m + p = N, the feed stage can be determined.

Summary of Equations and Correlations

• Minimum Reflux Ratio, $R_m$: Underwood Equation

• Minimum number of stages, $N_m$: Fenske Equation

• Gilliland Correlation, Erbar-Maddox Correlation

Example Problem

• Calculate:

  • (a) The minimum number of stages $N_m$ using the Fenske Equation.

  • (b) The minimum reflux ratio $R_m$ using the Underwood Equation.

  • (c) The theoretical number of stages, N by using the Erbar- Maddox correlation (given: R = 1.14$R_m$).

  • (d) The location of the feed point using the Kirkbride Equation.

Given Data

1. Identify heavy and light keys.

   • Heavy Key: iC5

   • Light Key: C4

2. Choose/Identify the reference component

   • If no $\alpha$ values given you can choose r to be any component you like.

   • If $\alpha$ values are given then use r as the component corresponding to $\alpha$ = 1.

   • In this case the reference component is iC5

3. Identify whether $\alpha$ varies significantly with T

   • In this case we will assume that $\alpha$ does not change significantly.

   • How could we investigate this in practice?

4. Identify component i

   • We will choose Butane – why?

   • Butane appears in both top and bottom streams

   • It also happens to be the light key

Example: Mole Fractions

LK: Butane
HK: Isopentane

Calculations and Tray Efficiency

• Fenske equation using number of moles instead of mole fractions (ratio of mole fractions).

• Fenske equation gives the theoretical number of stages.

• Actual number of trays is determined by the tray efficiency.

• Mathematically simpler to consider efficiency towards the end of the calculations.

Gilliland and Erbar-Maddox Correlations

• If $R<em>m$ is estimated as 1.185, and $R/R</em>m = 1.14$, estimate N using both the Gilliland and Erbar-Maddox correlations

• $R = (R/R<em>m) × R</em>m = 1.14 × 1.185 = 1.35$

• Remember, $N = 1 + n$, Since $N<em>m = 9.47$, therefore $n</em>m = 8.47$

• From above we get $n = 21.80$, and hence $N = 22.80$

Comparison of Correlations

• Gilliland Correlation gives 22.8 theoretical stages, Erbar-Maddox correlation gives 25.6

• Why are these values different?

  • Difficulty interpreting Erbar-Maddox correlation

  • Both methods are empirical, i.e. based on experimental data

  • Erbar-Maddox is generally considered to be the most accurate

Distribution of Non-Key Components

• We can predict the distribution of non-key components and thus revise our estimate of distillate and bottom products.

• Mean value for relative volatility is: $\alpha<em>{mean,av} = \sqrt{\alpha</em>{LK,av} \alpha_{HK,av}}$

Light components $(D<em>i > W</em>i)$

That is to say: $\alpha<em>{i,av} > \alpha</em>{mean,av}$
To revise the estimates of distillate products:
$W<em>i = \frac{F</em>i}{1 + V}$
$V=(\frac{D<em>i}{W</em>i})(\frac{W}{D})$

Heavy components $(W<em>i > D</em>i)$

That is to say: $\alpha<em>{i,av} < \alpha</em>{mean,av}$
To revise the estimates of bottom products:
$D<em>i = \frac{F</em>i}{1 + V}$
$V=(\frac{W<em>i}{D</em>i})(\frac{D}{W})$
Using the revised product distributions, verify the specifications.