Shortcut Distillation
Distillation is the most common unit operation for separating liquid mixtures into valuable and high purity products. The normal procedure for solving a typical multi-component distillation problem is to solve the MESH (Material balance, Equilibrium, Summation and Heat) balance equations stage-by-stage. Though computer programs are normally available for the rigorous solution of the MESH equations, short-cut methods are still useful in the preliminary design work, and as an aid in defining problems for computer simulation. This article describes a widely used shortcut distillation method commonly referred to as the Fenske-Underwood-Gilliland (FUG) method.
1. The components that have their distillate and bottoms fractional recoveries specified are called key elements. The most volatile of the keys is called the Light Key (LK) and the least volatile is called the heavy key (HK). The other components are called non-keys(NK). First step is to choose LK and HK and fix their distribution in top and bottom product.
2. Next step is to estimate overall top and bottom flow rate compositions assuming only light non-key components in top and heavy non-key component in bottoms as first attempt.
3. Estimate Dew point T_{Dew} for top composition and Bubble Point T_{Bubble} for bottom composition.
4. Estimate relative volatility with respect to the HK of all components at the T_{Dew} (top) and T_{Bubble} (bottom).
\displaystyle \displaystyle \alpha_{i}=K_{i}/K_{HK}
Calculate average volatilities for all components.
\displaystyle \displaystyle \alpha_{i,Avg}=\sqrt{\alpha_{i,Top}*\alpha_{i,Bottom}}
5. Recalculate overall top and bottom flow rate compositions on the basis of the Hengsteback and Geddes equation.
\displaystyle \displaystyle \log\left(\frac{d_{i}}{b_{i}}\right) = A + B\log(\alpha_{i})
Recovery of heavy key in bottoms is defined as b_{HK}/f_{HK}, recovery of light key in distillate is defined as d_{LK}/f_{LK}. Constants A & B are defined as:
\displaystyle \displaystyle A = \log\left(\frac{1-b_{HK}/f_{HK}}{b_{HK}/f_{HK}}\right)
\displaystyle \displaystyle B = \frac{\log\left( \frac{d_{LK}/f_{LK}}{1-d_{LK}/f_{LK}} * \frac{b_{HK}/f_{HK}}{1-b_{HK}/f_{HK}} \right)}{log\left(\alpha_{LK}\right)}
Recovery of i^{th} component in distillate is defined as:
\displaystyle \displaystyle \frac{d_{i}}{f_{i}} = \frac{10^{A}\alpha_{i}^{B}}{1+10^{A}\alpha_{i}^{B}}
Recovery of i^{th} component in bottom is defined as:
\displaystyle \displaystyle \frac{b_{i}}{f_{i}} = 1 - \frac{d_{i}}{f_{i}}
After calculating top and bottom compositions, go back to Step (3) and update T_{Dew} and T_{Bubble} and recalculate relative volatilites.
6. Estimate minimum number of stages, Nm using Fenske equation.
\displaystyle \displaystyle Nm = \frac{\log\left[ \left(\frac{x_{LK}}{x_{HK}}\right)_{d}*\left(\frac{x_{HK}}{x_{LK}}\right)_{b} \right]}{\log\left(\alpha_{LK}\right)}
where, (x_{LK}/x_{HK})_{d} is the ratio of mol fraction of light key and heavy key in distillate and (x _{HK}/x_{LK})_{b} is the ratio of mol fraction of heavy key and light key in bottoms.
7. Estimate minimum reflux ratio, Rm using Underwood's equation.
\displaystyle \displaystyle Rm+1=\sum\frac{\alpha_{i}x_{i,d}}{\alpha_{i}-\theta}
where Underwood constant, θ is calculated as the root of following equation:
\displaystyle \displaystyle 1-q=\sum\frac{\alpha_{i}x_{i,f}}{\alpha_{i}-\theta}
where, 1 < θ < α_{LK} and q is the quality of feed.
8. Select an operating reflux ratio, R as a factor of minimum reflux ratio, typically a factor of 1.2 is considered. Number of stages, N is calculated based on Gilliland's correlation.
\displaystyle \displaystyle X = \frac{R-Rm}{R+1}
\displaystyle \displaystyle Y = \frac{N-Nm}{N+1}
\displaystyle \displaystyle Y = 1-\exp\left[\left(\frac{1+54.4X}{11+117.2X}\right)\left(\frac{X-1}{X^{0.5}}\right)\right]
9. Feed stage location is estimated based on Kirkbride's equation.
\displaystyle \displaystyle \frac{m}{p}=\left[\left(\frac{B}{D}\right)\left(\frac{x_{HK}}{x_{LK}}\right)_{F}\left(\frac{x_{LK,B}}{x_{HK,D}}\right)^{2}\right]^{0.206}
\displaystyle \displaystyle n = m+p
where, m is number of stages above feed plate and p is number of stages below feed plate.
Resources
- Web based calculation available at checalc.com
- Spreadsheet for Shortcut Multi-Component Distillation based on FUG method
References
- Ludwig's Applied Process Design, Volume 2, 4^{th} edition, A. Kayode Coker
- An article on multi-component distillation by F.Grisafi