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`= Σ`

_{i}FXi/(1+θ.bd(i)_{ca}) - D

Above equation is solved using Newton Raphson method.

`g'(θ) = -Σ`_{i} bd(i)_{ca}.FXi/[1+θ.bd(i)_{ca}]²

New estimate of θ is made as following:

`θ`_{new} = 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} / Σ_{i} 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} / Σ_{i} 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.

Spreadsheet for Simple Multicomponent Distillation

### References

- Fundamentals of Multicomponent Distillation – C.D. Holland