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

### 2.2.1 The Enriching Operation Line

The enriching line is the same as was previously defined by Eq. 1-10 in part 1

Eq. 2-4) $\displaystyle \boxed{y=\frac{R_D}{R_D+1}x+\frac{x_D}{R_D+1}}$

### 2.2.2 The New q-Line

The q-line for stream $\displaystyle M$ is given by definition as

Eq. 2-5) $\displaystyle\boxed{y=\frac{q_M}{q_M-1}x-\frac{x_M}{q_M-1}}$

### 2.2.3 The Stripping Operation Line

The stripping operation line is the same as was previously defined by Eq. 1-45 in part 1

Eq. 2-6) $\displaystyle\boxed{y=\frac{\left(Fq_F+Gq_G+DR_D\right)x_Gx-\left(\left(F\left(q_F-q_G\right)+D\left(q_G+R_D\right)+q_GW\right)x+Wx_G\right)x_W+Wx^2_W}{\left(F\left(q_F-1\right)+G\left(q_G-1\right)+D\left(R_D+1\right)\right)x_G-\left(F\left(q_F-q_G\right)+D\left(q_G+R_D\right)+W\left(q_G-1\right)\right)x_W}}$

## 2.3.0 Solving the Initial Problem

At this point, the solution method is the trivial case of a column with a single feed and a defined enriching operation line.  Rather than waste time here, though I may come back to it later, I will instead refer to the Wikipedia article, which does a decent enough job given the more complicated construction already demonstrated in part 1.

Part 3 is on the way!  It is the real meat of the proof where I show that both methods are equivalent using some simple geometry and a ratio test.

Be Sociable, Share!
Short URL: http://dlvr.it/1BBGd7