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 15, 2012  Tagged with:  Comments Off on Useful Definitions and Identities for Thermodynamic Potentials

This is just a summary of many of the definitions and identities I find useful when working with the basic thermodynamics of a system or process.  It is not meant to be a comprehensive list.

# Mathematical Identities

Using the reasoning behind the notation used for derivatives of thermodynamic potentials, there are some useful identities given the following prototypical system

Eq. 1) $\displaystyle X=X\left(Y,Z\right)$

This equation represents a generic thermodynamic potential of an arbitrary system or process as a function of two implicitly extensive thermodynamic potentials.  Adopting the notation

Eq. 2) $\displaystyle\underline{X}=\frac{X}{N}$

To denote the relationship between intensive variables, extensive variables, and the size of the system allows for the trivial conversion between system size dependent and system size agnostic thermodynamic relationships.  The total differential for the extensive form of the thermodynamic potential for the prototypical system is given by

February 14, 2012  Tagged with: , ,

When I took my process thermodynamics course as an undergraduate student, I was told to use what I had initially considered to be a completely redundant notation for partial derivatives for thermodynamic potentials.  The notation involved wrapping the partial derivative in a set of parentheses and noting which variables were “held constant” as a subscript.

Ex. $\displaystyle\left(\frac{\partial P}{\partial V}\right)_{T,\vec{N}}$

The derivation leading to this was woefully devoid of the mathematical basis for this apparently redundant notation.  Furthermore, so was the general literature on the subject, most of which consisted of fleeting introductions to their respective application.  As I played with the idea, it became clear why this notation was indeed very necessary.  It was only as I was solving a separate problem on my own that such notation yielded valuable information about the nature of the potentials being described.

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

February 7, 2012  Tagged with: , , , ,  Comments Off on A Great Tool for Visualizing Disk Space

I’ve been using this free tool called “Folder Size” (download available here) for a while now. It is a very impressive piece of (free) software that scans the entirety of a selected drive and produces an empirical picture of the distribution of occupied drive space. I have found this to be to be invaluable when cleaning out my disk space as it allows me to see the effective size of folders as well as files. The ability to natively view the size of a folder in explorer has been missing from MS Windows since its inception. Originally, I stumbled upon this application while attempting to address that specific concern, which it does. Beyond displaying the size of folders, it also shows the relative sizes of all items at the same branch of the drive hierarchy in both a percent column in the analysis results and a graphical display below as well as detailed file and folder attributes. Some of the options for the data displayed are type of visualization and default size basis (think KB, MB, GB, etc.). I highly recommend this software if you are going through the “where did all my disk space go?!?!?” routine.

February 1, 2012  Tagged with: , , , , ,  Comments Off on I am Now a Certified Engineer in Training in Florida

I decided to swing by the Florida license website to check my application status and discovered that I’m now certified as an engineer in training (EIT). This is really exciting! Now I just need to fulfill the experience requirement and take the principles and practices of engineering exam to be certified as a professional engineer (PE). The hope is to end up working in a state that allows for submission of an awarded higher degree (such as the Ph.D. I will be pursuing later this year) in lieu of years of work experience so that I may begin the process for the PE license sooner rather than later. 