## Simple Multicomponent Distillation

This article shows application of Thiele and Geddes method to solve simple multi-component distillation problem in excel spreadsheet based on θ method of convergence.

Antoine equation is used to calculate equilibrium constant Ki.

``Ki = exp(Ai - Bi / (T + Ci))/P``

where, Ai, Bi & Ci are Antoine equation constants for a component i, T & P are temperature and pressure at a stage.

### Feed Analysis

To start with bubble point pressure (PBubble) and dew point pressure (PDew) are determined for feed mixture.

• P < PDew, Mixture exists as superheated vapor.
• P > PBubble, Mixture exists as sub-cooled liquid.
• PDew < P < PBubble, mixture exist in vapor and liquid phase.

For last case, PT Flash calculation is performed to determine liquid (Lf) & vapor (Vf) fraction and molar composition.

#### Stage-wise Calculation

Liquid and Vapor flow across column are determined as:

````L (Above Feed Stage) = R.D`
`V (Above Feed Stage) = (1+R).D`
`L (Below Feed Stage) = R.D + Lf`
`V (Below Feed Stage) = (1+R).D - Vf````

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Start of Iteration
A Linear temperature gradient is assumed across the column. Ki values are calculated for all component on each stage based on Antoine equation.

θ Method of Convergence
L/D ratios are calculated for all components from top to stage above feed stage.

Top Stage
Total Condenser

``LD(1,i) = L(1)/D``

Partial Condenser

``LD(1,i) = L(1)/(K(1,i).D)``

where 1 indicates top stage, i denotes the component and LD denotes L/D ratio.

Above Feed Stage

``LD(j,i) = (LD(j-1,i) + 1).L(j)/(K(j,i).V(j))``

where j represents the stage and i represents a single component.

Similarly L/B ratios are calculated from bottom upto feed stage

Bottom Reboiler

``LB(bottom) = 1``

L/B ratios from bottom upto feed tray is calculated as:

``LB(j,i) = LB(j+1,i).K(j+1,i).V(j+1)/L(j+1)  + 1 ``

For Feed Stage

``Vfb(i) = V(f).K(f,i)/( L(f).LB(f,i) )``

B/D ratio for feed plate is calculated as:

``bd(i) = ( LD(nf-1,i) + Lf.Xfi/(F.Zfi))/(Vfb(i) + Vf.Yfi/(F.Zfi))``

where bd represent B/D ratio at feed plate.

Based on overall component material balance.

````F.Xi = D.Xdi + B.Xbi`
`d(i) = D.Xdi`
`b(i) = B.Xbi`
`F.Xi = d(i) + b(i)`
`Σd(i) = D````

A multiplier θ is defined as:

``[b(i)/d(i)]corrected = θ[b(i)/d(i)]calculated``

Above equation is rewritten as:

``bd(i)co = θ.bd(i)ca``

Function g(θ) is defined as:

````d(i)co = FXi/(1+θ.bd(i)ca)`
`g(θ)  = Σd(i)co - D`
`      = Σi FXi/(1+θ.bd(i)ca) - D````

Above equation is solved using Newton Raphson method.

``g'(θ) = -Σi bd(i)ca.FXi/[1+θ.bd(i)ca]²``

New estimate of θ is made as following:

``θnew = θ - g(θ)/g'(θ)``

Iteration is done till there is negligible change in value of θ. Converged value of θ is used to estimate d(i)co. Value of b(i)co is calculated as following:

``b(i)co = θ.d(i)co.bd(i)ca``

Liquid mol fraction from top to stage above feed stage are calculated:

``X(j,i) = LD(j,i)ca.d(i)co / Σi LD(j,i)ca.d(i)co``

Liquid mol fraction from feed to bottom tray are calculated:

``X(j,i) = LB(j,i)ca.b(i)co / Σi LB(j,i)ca.b(i)co``

Bubble point temperature is calculated for each stage. With this first iteration completes. In next iteration new set of Ki values are calculated for each tray based on updated temperature profile and all of the above steps are repeated till there is no change in temperature profile and θ value becomes 1.

### References

• Fundamentals of Multicomponent Distillation – C.D. Holland

## Binary Batch Distillation

Batch distillation is widely used in chemical processing industries when high value added, low volume chemicals must be separated. This article shows how to model and solve a multi-stage binary batch distillation problem in excel.

Example
The 50 mol feed comprising 70% mols of A and 30% mols of B is to be distilled in the multi-stage batch distillation with 5 equilibrium stages on top of the reboiler (still pot). Reflux is returned to the column as a saturated liquid with the constant reflux ratio of 1.5. Relative volatility of A is 1.8. It is desired to reduce the mols of A in still to 10%. Determine average distillate composition, final amount of liquid in still pot and total amount of distillate collected.

A material balance is done across the column and following relationship commonly known as Rayleigh Equation is derived.

Where F (Initial mols fed in still), WF (Mols left in still at any time), xF (Initial mol fraction in feed), xW (Mol fraction in still at any time) and xD (Distillate composition at any particular time).

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Equilibrium curve is obtained from relative volatility.

`` y = αx / (1+ x (α - 1))``

Operating line is obtained as following.

`` y = Rx /(R+1) + xD /(R+1)``

Relationship between xD and xW is obtained by performing stage-by-stage calculations. A value of xD is selected and stages are stepped off (for a given number of equilibrium stages) to find the value of xW. It is done for all possible values of xD ranging from 0 to 1. A plot of 1/(xD – xW) vs xW is made.

Area under curve is calculated from xW to xF by calculating area of trapezoid formed in small intervals.

`` Area (Cell P59) = (N60-N59)*(O60+O59)/2)``

Area of all trapezoids in the range from xW to xF are added to get area under curve.

`` Area = 2.153``

Amount left in still, WF is calculated as following.

```` WF = F.exp(- Area under the curve)`
` WF = 5.80 mols````

Distillate mols collected is obtained as following.

```` D = F - WF`
` D = 44.20 mols````

Average composition of distillate collected xD is calculated as following.

```` xD = (xF F - xW WF)/D`
` xD = 0.779 ````

Spreadsheet for Multi-stage Binary Batch Distillation

## McCabe Thiele Diagram

This article shows how to draw McCabe Thiele diagram for doing binary distillation analysis in a spreadsheet.

Example
Determine total number of stages and feed stage for separation of a binary mixture with relative volatility (α) 2.5, Feed composition (zF) 0.36, Distillate Composition (xD) 0.915, Bottom composition (xB) 0.05 and feed quality (q) 1.5. Reflux ratio to be 1.5 times the minimum reflux ratio (Rm).

### Minimum Reflux Ratio (Rm)

Calculate the intersection of q-Line and equilibrium curve.

```` y = qx /(q-1) - zF /(q-1)`
` y = αx / (1+ x (α - 1))````

Solving above equations gives a quadratic expression, which is solved for positive roots.

```` x² (q(α-1)) + x (q - zF (α -1) - α(q-1)) - zF = 0`
` x1 = 0.470`
` y1 = 0.689````

Draw a line from ( xD, yD) passing through ( x1, y1) with an intercept xD / (Rm + 1) on Y axis.

```` xD / (Rm + 1) = 0.450`
` Rm = 1.032````

Reflux ratio is calculated as 1.5 times the minimum reflux ratio.

```` R = 1.5 x Rm`
` R = 1.55````

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### Operating Lines

Intersection of q-Line and rectification section operating line is calculated.

```` y = qx /(q-1) - zF /(q-1)`
` y = Rx /(R+1) + xD /(R+1)`
` x2 = 0.451`
` y2 = 0.633````

Draw rectification section operating line from (xD, yD) to (x2, y2) and stripping section operating line from (x2, y2) to (xB, yB).

### Draw Stages

Start drawing steps from (xD, yD). Draw a horizontal line by calculating x value from equilibrium curve based on yD. Check whether this x value is greater than x2. As long as it is greater than x2 rectification section operating line is used for calculation of y value. For next step this new value of y is used and rest of the steps are repeated. If x becomes less than x2 stripping section operating line is used for calculation of y value. It is continued till it is greater than xB.

Feed stage is determined by checking at each stage when x value becomes less than x2, similarly total number of stages are determined by checking at each stage when x value become less than xB. At last stage value of x is calculated and difference with xB is added to give total number of stages.

```` Feed Stage   = 5`
` Total Stages = 11.26````

Spreadsheet for Binary Distillation analysis based on McCabe Thiele method