May 1, 2012  Tagged with: , , , , , , , ,  Comments Off on Distillation of a Binary Mixture in a Distillation Column of Arbitrary Design, Part 5

*Note: This article assumes you have rudimentary knowledge of how a simple distillation column operates. Though much of the basics of the simple model for a distillation column will be covered below, it should not be considered as a stand alone reference.

# 5.0 Conclusions

Since there were several motivations for this derivation, the conclusions that may be drawn from it overall will be broken down into two relevant categories for discussion and summary.

February 29, 2012  Tagged with: , , , , , , , ,  Comments Off on Distillation of a Binary Mixture in a Distillation Column of Arbitrary Design, Part 4

*Note: This article assumes you have rudimentary knowledge of how a simple distillation column operates. Though much of the basics of the simple model for a distillation column will be covered below, it should not be considered as a stand alone reference.

# 4.0 Generalization to Columns of Arbitrary Design

## 4.1.0 Motivation for Generalizing to an Arbitrary Design

The motivation to generalize to a system with an arbitrary topology is simply to take the next logical step and expand on what has already been accomplished. The case for two feed/side streams was itself an extension of the simple case of a single feed stream.

## 4.2.0 Physical Description for a Generalized Column Model

To generalize to an arbitrary topology, some choices as far as the idealized general topology have to be made. This leads to the least complex yet most regularly structured limit of including a single feed stream and a single side stream placed optimally at each theoretical plate. So, the column is now comprised of $\displaystyle n$ plates, each of which is associated with a feed stream ( $\displaystyle F_i$, $\displaystyle x_{F_i}$, and $\displaystyle q_{F_i}$), a side draw stream ( $\displaystyle S_i$, $\displaystyle x_{S_i}$, and $\displaystyle q_{S_i}$), a liquid overflow entering from above ( $\displaystyle L_{i-1}$ and $\displaystyle x_{i-1}$) and underflow leaving below ( $\displaystyle L_i$ and $\displaystyle x_i$), a vapor overflow leaving above ( $\displaystyle V_i$ and $\displaystyle y_i$) and underflow entering from below ( $\displaystyle V_{i+1}$ and $\displaystyle y_{i+1}$), and the distillate and waste streams with their associated values as described in part 1.

## 4.3.0 Method of Model Generalization

February 10, 2012  Tagged with: , , , , , , , ,  Comments Off on Distillation of a Binary Mixture in a Distillation Column of Arbitrary Design, Part 3

*Note: This article assumes you have rudimentary knowledge of how a simple distillation column operates. Though much of the basics of the simple model for a distillation column will be covered below, it should not be considered as a stand alone reference.

# 3.0 Proof of the Equivalence of Methods 1 and 2

## 3.1.0 Geometric Basis for Equivalence

### 3.1.1 The Rationale of the Method for Comparison

Given that method 1 and method 2 involve plotting the exact same enriching and stripping operation lines, the enriching and stripping operation lines are by definition not parallel for any real physical system that does not involve an azeotrope**, and the combined streams q-line from method 2 also passes through the intersection of said enriching and stripping operation lines, there exists at the point of intersection of these three lines a set of three different means of determining the coordinates of the point based on the intersection of a pair of lines.  Two of these lines, the enriching and stripping operation lines, are present in both methods and are defined by Eq. 1-10 and Eq. 1-45 in method 1.  The third line, the combined streams q-line, is defined by Eq. 2-5 in method 2.  As a matter of geometric consistency, evaluation of the ratio of x-ordinates or y-ordinates for a pair of such intersections should result in unity.  In other words, when taking the ratio of a single ordinate between a pair of definitions for the intersection of the three lines, all variables should cancel exactly.

### 3.1.2 The Points to be Examined

The pair of definitions for the intersection of the three lines will be taken as the intersection of the enriching and stripping operation lines and the intersection of the enriching operation line and the combined streams q-line.  The latter selection was made due to the relative simplicity of the definition of the enriching operation line over that of the stripping operation line.

## 3.2.0 Proof of Equivalence

February 9, 2012  Tagged with: , , , , , , , ,

*Note: This article assumes you have rudimentary knowledge of how a simple distillation column operates. Though much of the basics of the simple model for a distillation column will be covered below, it should not be considered as a stand alone reference.

# 2.0 Solution Method 2: The Combined Streams Modification of the McCabe-Thiele Method

## 2.1.0 Defining the Combined Feed/Side Stream

The new stream will be defined as a combined side stream with flow rate $\displaystyle M$, composition $\displaystyle x_M$, and fractional quality $\displaystyle q_M$.

### 2.1.1 The New Flow Rate

The flow rate for the combined side stream is defined as the following sum

Eq. 2-1) $\displaystyle\boxed{M=F+G}$

### 2.1.2 The New Composition

The composition for the combined side stream is defined as the following weighted average

Eq. 2-2) $\displaystyle\boxed{x_M=\frac{Fx_F}{M}+\frac{Gx_G}{M}}$

This is a classic application of the lever rule.

### 2.1.3 The New Fractional Quality

The fractional quality of the combined side stream is defined as the following weighted average

Eq. 2-3) $\displaystyle\boxed{q_M=\frac{Fq_F}{M}+\frac{Gq_G}{M}}$

## 2.2.0 Defining Relevant Operation Lines and q-Lines

January 31, 2012  Tagged with: , , , , , , , ,

*Note: This article assumes you have rudimentary knowledge of how a simple distillation column operates.  Though much of the basics of the simple model for a distillation column will be covered below, it should not be considered as a stand alone reference.

# Introduction

While taking my separations and mass transfer operations course at UF, I took a particularly interesting exam where I was tasked with predicting the performance of a distillation column with two inputs.  This may be getting ahead of myself, as the problem’s information was written into the proof, but the problem was as follows:

## The Test Problem

Given a distillation column separating n-pentane and n-heptane, with a feed rate of 200 kgmol/hr of a 40 mol% n-pentane liquid at bubble point, a 95 mol% distillate stream, a 5 mol% bottoms stream, a 30 mol% side stream with a flow rate out equal to that of the bottoms leaving, a reflux ratio equal to twice that of the minimum, and a 50% average tray efficiency, use the McCabe-Thiele graphical method and the provided vapor-liquid equilibrium data to determine a) the flow rate and composition of all streams, b) the minimum reflux ratio, c) the number of theoretical plates required, and d) the optimum placement of the feed stream and side stream.

So, the information given so far is

• $F=200 kgmol/hr$
• $x_F=0.40$
• $q_F=1$
• $x_D=0.95$
• $x_W=0.05$
• $G=-W$ (negative sign due to opposite orientation)
• $x_G=0.30$
• $q_G=1$
• $R_D=2R_{min}$
• $\epsilon_a=0.50$

January 16, 2012  Comments Off on Old Work to be Posted Over the Next Few Months Over the next few months I will be posting about many of the independent projects and research projects I did as an undergraduate.  They range from metabolic models to full process designs and cover a diverse selection of chemical engineering principles, several with significant real world applications.  Starting things off will be a walk through of a proof I wrote in response to a test grade I did not agree with and how the results of the proof may be used both to simplify and enhance a graphical method for predicting distillation column behavior.  This proof will be the basis for a paper I will be writing on the subject as well.