3G4 Distillation Calculations

DISTILLATION: McCABE-THIELE DIAGRAMS AND SHORTCUT METHODSChE 3G4 Spreadsheet

Distillation columns can typically be described by the schematic diagram shown to the right.

In designing a column, we can identify two practical limiting cases for the reflux ratios L/V and L/D:

A column contains N trays, each of which is at a particular temperature and pressure. Vapour-liquid equilibrium is established across each of these trays, with a vapour flow rate of V and liquid flow rate of L. A feed stream of molar flow F, mole fraction composition zi and quality q (q=0 is a saturated vapour; q=1 is a saturated liquid) is fed to the column at an optimized tray NFEED. The vapour from the top of the column (molar flow rate V) is totally condensed, with part of the condensate returned to the column (molar flow rate L) and part removed as a distillate product (molar flow rate D with mole fraction composition xi,DIST). Similarly, the liquid from the bottom of the column is partially reboiled back to the column, with the remaining liquid portion removed as a bottoms product (molar flow rate B with a mole fraction composition xi,BOT). Unlike in a flash drum, the product distillate and bottoms streams are NOT themselves in equilibrium, only the vapour and liquid compositions of each single tray.

Distillation processes are frequently used in industry to do perform well-defined separations, often using a cascade of columns in sequence to achieve the desired product compositions. As a result, it's important we find a way to design columns to meet specific product stream specifications. A variety of methods can be used, the most obvious of which is performing tray-by-tray balances within the column. Since each tray is at a constant pressure, this essentially amounts to performing a flash calculation on each single tray up and down the column. While computer simulation programs can do such calculations, they are very cumbersome and difficult to do with a spreadsheet. However, we can use shortcut methods to estimate the number of trays and external (L/D) and internal (L/V) reflux ratios required to produce product streams of specified compositions.

(1) L=V; that is, we do not take any distillate (or bottoms) product

In this case, we need a minimum number of trays to perform the separation since we have no incoming or outgoing flow. This minimum number of trays (Nmin) can be predicted using the Fenske equation, expressed either in terms of the mole fractions (x) of components A and B or the fractional recoveries (FR) of components A and B in the distillate and bottom stream. Here, A is the light key component (ie. a component present in both the distillate and bottoms but recovered primarily in the distillate) and B is the heavy key component (ie. a component present in both the distillate and bottoms but recovered primarily in the bottoms). The light and heavy keys are the two components whose recovery in the distillate and/or bottoms is specified in the problem in a multi-component distillation problem:

The parameter aAB in these equations is the relative volatility of A with respect to B, which (assuming ideal conditions) can be estimated as:

where PA* and PB* are the vapour pressures of components A and B (from Antoine's equation)

Nmin=

ln { (FRA)DIST (FRB)BOT[1−(FRA)DIST ] [1−(FRB)BOT ] }

ln α AB

Nmin=

ln {( xA

xB )DIST /(xA

xB )BOT }ln α AB

α AB=

yA

xA

yBxB

≈

P¿A

P

P¿B

P

=PA

¿

PB¿

(1) (2)

A key assumption to this approach is that the relative volatility is constant throught the entire column, despite the fact that a range of temperatures are present on the different trays. Several approaches are taken to get the "best" estimate of this average relative volatility; in this spreadsheet, the dew and bubble temperatures are calculated and the partial pressure ratio at the midpoint of these temperatures is used.

(1) L/Dmin - the external reflux ratio at which the specified separation is just achieved with an infinite number of trays

As we continue to take more and more product off the top, we reduce the amount of product returned to the column and consequently reduce the total time an average molecule stays within the column. Eventually, we reach a point where an "infinite" number of stages is required to separate the components according to the column specifications. This minimum external reflux ratio L/Dmin can be predicted using the Underwood equations given below:

To solve these equations, we first find the value of the parameter f which satisfies equation (1). We can then substitute this value into (2) to calculate Vmin. D can be calculated based on the feed flow and composition and the specified mole fractions and/or fractional recoveries of the components. Lmin can be calculated using equation 3. These equations can be used for any number of different components within the column; however, if solved in this fashion (ie. with a single value of f), it is required to assume that any light or heavy non-key components (ie. components with relative volatilities higher and lower than the light and heavy key components respectively) either do not distribute at all (ie. all light non-key ends up in the distillate and all heavy non-key in the bottoms) or distribute according to the Fenske equation prediction:

We can then use these extreme results (ie. the minimum possible number of trays and the minimum possible external reflux ratio) to predict the number of trays and reflux ratio required in a real column using the Gilliland correlation. A design value for L/D is first specified, usually as some factor of (L/D)min (typical values are between 1.05 and 1.25). The following values are then calculated and fit to the correlation developed by Gilliland:

We can therefore solve for N as well as the optimum feed location in a real column, based on the N/Nmin scaling and the Fenske optimum feed prediction:

Practically, the Fenske-Underwood-Gilliland approach gives rough, first-pass estimates of the number of stages required to perform a given separation. However, the assumption of constant relative volatility can be inaccurate in some cases, particularly in columns with highly non-ideal components and/or a large temperature range.

ΔV FEED=∑i

αi Fziα i−φ

=F (1−q ) V min=∑i

αiDxi ,DISTαi−φ

Lmin=Vmin−D

N F ,min=

ln {( xAxB )DIST /(zAzB )}

lnα AB

N F=N F ,min( NNmin )

x=Abscissa=

LD

−(LD)min

LD

+1

α AB=

yA

xA

yBxB

≈

P¿A

P

P¿B

P

=PA

¿

PB¿

(FRC )DIST=αCB

N min

(FRB)BOT1−(FRB )BOT

+αCB

Nmin

y=N−Nmin

N+1

Feed Line:

Top Operating Line:

Bottom Operating Line:

We also plot the simple y=x line on the graph. A typical McCabe-Thiele plot is shown below (with the polynomial y vs x VLE fit)

This spreadsheet will allow you to use both the Fenske-Underwood-Gilliland and the McCabe-Thiele approach to design distillation columns.

Stage-by-stage calculations can be performed graphically using the McCabe-Thiele method. In this approach, the y vs x VLE relationship is plotted directly on the graph, eliminating the uncertainty regarding the constant relative volatility estimate. This means we need to perform dew or bubble point calculations to generate the y vs x equilibrium data; in this spreadsheet, we use an ideal bubble point calculation to produce this data. The curve is then fit to a fourth-order polynomial expression in order to give us an algebraic expression for the y vs x equilibrium line, allowing us to calculate its intercepts with the other lines. On the same graph, we plot three additional lines:

Starting at the specified mole fraction of component A (the lighter component) in the distillate on the y=x line, we can step down the curves, using the y vs. x VLE equilibrium data curve and either the top or bottom operating lines as our step limits. The top operating line is used when the component A mole fraction is greater than the x value of the feed line - y vs x VLE curve intercept; the bottom operating line is used at mole fractions below the intercept. This method can be used to design any column with any specifications (ie. we are not limited to total reflux or minimum reflux). However, by setting L/V equal to one, we can use the McCabe-Thiele diagram to check that the Fenske calculation of Nmin is accurate. In the case of minimum reflux, we can obviously not plot an infinite number of stages using the McCabe-Thiele method; however a "pinch point" will be visible on the graph in which the operating and equilibrium lines touch.

yA=LV

xA+(1− LV ) xA ,DIST

yA=q

q−1xA+

11−q

zA ,FEED

yA=LV

xA+(1− LV ) xA ,BOT

Cells highlighed in YELLOW require input from youCells highlighted in BLUE require you to perform a manual GoalSeek procedure on that cell to get a converged solution

Cells with a red triangle in the upper right-hand corner have comments which will give you more information about what the variable in the cell means or how to select a value for that variable.

where

In this case, we need a minimum number of trays to perform the separation since we have no incoming or outgoing flow. This minimum number of trays (Nmin) can be , expressed either in terms of the mole fractions (x) of components A and B or the fractional recoveries (FR) of components A and

B in the distillate and bottom stream. Here, A is the light key component (ie. a component present in both the distillate and bottoms but recovered primarily in the distillate) and B is the heavy key component (ie. a component present in both the distillate and bottoms but recovered primarily in the bottoms). The light and heavy keys are the two components whose recovery in the distillate and/or bottoms is specified in the problem in a multi-component distillation problem:

in these equations is the relative volatility of A with respect to B, which (assuming ideal conditions) can be estimated as:

* are the vapour pressures of components A and B (from Antoine's equation)

Nmin=

ln { (FRA)DIST (FRB)BOT[1−(FRA)DIST ] [1−(FRB)BOT ] }

ln α AB

(FR A )DIST=DxA ,DIST

Fz A

(FRB )BOT=Bx A ,BOT

Fz A

B

DV

L

N

V

L

Fzi

q

xi,BOT

xi,DIST

1

NFEED

(3)

A key assumption to this approach is that the relative volatility is constant throught the entire column, despite the fact that a range of temperatures are present on the different trays. Several approaches are taken to get the "best" estimate of this average relative volatility; in this spreadsheet, the dew and bubble temperatures are

- the external reflux ratio at which the specified separation is just achieved with an infinite number of trays

As we continue to take more and more product off the top, we reduce the amount of product returned to the column and consequently reduce the total time an average molecule stays within the column. Eventually, we reach a point where an "infinite" number of stages is required to separate the components according to the column

Underwood equations given below:

which satisfies equation (1). We can then substitute this value into (2) to calculate Vmin. D can be calculated based on the feed flow and composition and the specified mole fractions and/or fractional recoveries of the components. Lmin can be calculated using equation 3. These equations can be used for any number of different components within the column; however, if solved in this fashion (ie. with a single value of f), it is required to assume that any light or heavy non-key components (ie. components with relative volatilities higher and lower than the light and heavy key components respectively) either do not distribute at all (ie. all light non-key ends up in the distillate and all heavy non-key in the bottoms) or distribute according to the Fenske

We can then use these extreme results (ie. the minimum possible number of trays and the minimum possible external reflux ratio) to predict the number of trays and reflux ratio required in a real column using the Gilliland correlation. A design value for L/D is first specified, usually as some factor of (L/D)min (typical values are between 1.05 and 1.25). The following values are then calculated and fit to the correlation developed by Gilliland:

We can therefore solve for N as well as the optimum feed location in a real column, based on the N/Nmin scaling and the Fenske optimum feed prediction:

Practically, the Fenske-Underwood-Gilliland approach gives rough, first-pass estimates of the number of stages required to perform a given separation. However, the assumption of constant relative volatility can be inaccurate in some cases, particularly in columns with highly non-ideal components and/or a large temperature range.

Lmin=Vmin−D

We also plot the simple y=x line on the graph. A typical McCabe-Thiele plot is shown below (with the polynomial y vs x VLE fit)

This spreadsheet will allow you to use both the Fenske-Underwood-Gilliland and the McCabe-Thiele approach to design distillation columns.

Stage-by-stage calculations can be performed graphically using the McCabe-Thiele method. In this approach, the y vs x VLE relationship is plotted directly on the graph, eliminating the uncertainty regarding the constant relative volatility estimate. This means we need to perform dew or bubble point calculations to generate the y vs x equilibrium data; in this spreadsheet, we use an ideal bubble point calculation to produce this data. The curve is then fit to a fourth-order polynomial expression in order to give us an algebraic expression for the y vs x equilibrium line, allowing us to calculate its intercepts with the other lines. On the same graph, we plot three additional

Starting at the specified mole fraction of component A (the lighter component) in the distillate on the y=x line, we can step down the curves, using the y vs. x VLE equilibrium data curve and either the top or bottom operating lines as our step limits. The top operating line is used when the component A mole fraction is greater than the x value of the feed line - y vs x VLE curve intercept; the bottom operating line is used at mole fractions below the intercept. This method can be used to design any column with any specifications (ie. we are not limited to total reflux or minimum reflux). However, by setting L/V equal to one, we can use the McCabe-Thiele diagram to

is accurate. In the case of minimum reflux, we can obviously not plot an infinite number of stages using the McCabe-Thiele method; however a "pinch point" will be visible on the graph in which the operating and equilibrium lines touch.

Cells highlighed in YELLOW require input from youCells highlighted in BLUE require you to perform a manual GoalSeek procedure on that cell to get a converged solution

Cells with a red triangle in the upper right-hand corner have comments which will give you more information about what the variable in the cell means or how to select a

ANTOINE EQUATION COEFFICIENTS

Use the numbers in column B to choose your components in the subsequent spreadsheets

No. Substance Formula Range (ºC) A B C

1 Acetaldehyde -0.2 to 34.4 8.00552 1600.017 291.809

2 Acetic Acid 29.8 to 126.5 7.38782 1533.313 222.309

3 Acetic Acid 0 to 36 7.18807 1416.700 225.000

4 Acetic Anhydride 62.8 to 139.4 7.14948 1444.718 199.817

5 Acetone -12.9 to 55.3 7.11714 1210.595 229.664

6 Acrylic Acid 5.65204 648.629 154.683

7 Ammonia 7.55466 1002.711 247.885

8 Aniline 7.32010 1731.515 206.049

9 Benzene 6.89272 1203.531 219.888

10 n-Butane 6.82485 943.453 239.711

11 i-Butane 6.78866 899.617 241.942

12 1-Butanol 7.36366 1305.198 173.427

13 2-Butanol 7.20131 1157.000 168.279

14 1-Butene 6.53101 810.261 228.066

15 Butyric Acid 8.71019 2433.014 255.189

16 Carbon disulfide 6.94279 1169.110 241.593

17 Carbon tetrachloride 6.87926 1212.021 226.409

18 Cholorobenzene 0 to 42 7.10690 1500.000 224.000

19 Cholorobenzene 42 to 230 6.94504 1413.120 216.000

20 Chloroform -30 to 150 6.90328 1163.030 227.400

21 Cumene 6.93619 1460.310 207.701

22 Cyclohexane 6.84941 1206.001 223.148

23 Cyclohexanol 6.25530 912.866 226.232

24 n-Decane 6.95707 1503.568 194.73825 1-Decene 6.95433 1497.527 197.05626 1,1-Dicholoroethane 6.97702 1174.022 229.06027 1,2-Dicholoroethane 7.02530 1271.254 222.92728 Dicholoromethane 7.40916 1325.938 252.61629 Diethyl ether 6.92032 1064.066 228.79930 Diethyl ketone 7.02529 1310.281 214.19231 Dimethylamine 7.08212 860.242 221.66732 N,N-Dimethylformamide 6.92796 1400.869 196.43433 1,4-Dioxane 7.43155 1554.679 240.33734 Ethanol 19.6 to 93.4 8.11220 1592.864 226.18435 Ethanolamine 7.45680 1577.670 173.36836 Ethyl acetate -20 to 150 7.09808 1238.710 217.00037 Ethyl chloride 6.98647 1030.007 238.61238 Ethylbenzene 56.5 to 137.1 6.95650 1423.543 213.091

C2H4O

C2H4O2

C2H4O2

C4H6O3

C3H6O

C3H4O2

NH3

C6H7N

C6H6

n-C4H10

i-C4H10

C4H10O

C4H10O

C4H8

C4H8O2

CS2

CCl4

C6H5Cl

C6H5Cl

CHCl3

C9H12

C6H12

C6H12O

n-C10H22

P¿=10(A− B

T−C )

39 Ethylene glycol 8.09083 2088.936 203.45440 Ethylene oxide 8.69016 2005.779 334.76541 1,2-Ethylenediamine 7.16871 1336.235 194.36642 Formaldehyde 7.19578 970.595 244.12443 Formic Acid 7.58178 1699.173 260.71444 Glycerol 6.16501 1036.056 28.09745 n-Heptane 6.90253 1267.828 216.82346 6.87689 1238.122 219.78347 n-Hexane 6.88555 1175.817 224.86748 6.86839 1151.041 228.47749 Hydrogen Cyanide 7.52823 1329.490 260.41850 Methanol -20 to 140 7.87863 1473.110 230.00051 Methyl acetate 7.06524 1157.630 219.72652 Methyl bromide 7.09084 1046.066 244.91453 Methyl choride 7.09349 948.582 249.33654 Methyl ethyl ketone 7.06356 1261.339 221.96955 Methyl isobutyl ketone 6.67272 1168.408 191.94456 Methyl methacrylate 8.40919 2050.467 274.36957 Methylamine 7.33690 1011.532 233.28658 Methylcyclohexane 6.82827 1273.673 221.72359 Naphthalene 7.03358 1756.328 204.84260 Nitrobenzene 7.11562 1746.586 201.78361 Nitromethane 7.28166 1446.937 227.60062 n-Nonane 6.93764 1430.459 201.80863 1-Nonane 6.95777 1437.862 205.81464 n-Octane 6.91874 1351.756 209.10065 6.88814 1319.529 211.62566 1-Octene 6.93637 1355.779 213.02267 n-Pentane 6.84471 1060.793 231.54168 6.73457 992.019 229.56469 1-Pentanol 7.18246 1287.625 161.33070 1-Pentene 6.84268 1043.206 233.34471 Phenol 7.13301 1516.790 174.95472 1-Propanol 7.74416 1437.686 198.46373 2-Propanol 7.74021 1359.517 197.52774 Propionic acid 7.71423 1733.418 217.72475 Propylene oxide 7.01443 1086.369 228.59476 Pyridine 7.04115 1373.799 214.97977 Styrene 7.06623 1507.434 214.98578 Toluene 6.95805 1346.773 219.69379 1,1,1-Trichloroethane 8.64344 2136.621 302.76980 1,1,2-Trichloroethane 6.95185 1314.410 209.19781 Trichloroethylene 6.95185 1314.410 209.19782 Vinyl acetate 7.21010 1296.130 226.655

83 0 to 60 8.10765 1750.286 235.000

84 60 to 150 7.96681 1668.210 228.00085 7.00646 1460.183 214.82786 7.00154 1476.393 213.872

i-Heptane

i-Hexane

i-Octane

i-Pentane

Water1

Water2

m-Xyleneo-Xylene

87 6.98820 1451.792 215.110p-Xylene

Use the numbers in column B to choose your components in the subsequent spreadsheets

TWO-COMPONENT DISTILLATION: ESTIMATION OF RELATIVE VOLATILITY

Parameters for Chosen Set of Components

Component A 9 Benzene Component B 78 Toluene

Antoine's Equation Coefficients for Benzene Antoine's Equation Coefficients for TolueneA= 6.89272 A= 6.95805B= 1203.531 B= 1346.773C= 219.888 C= 219.693

0.5 Benzene 0.5 Toluene

Partial Pressure Temperature = 50.000 ˚C Partial Pressure Temperature = 50.000

271.236 mm Hg 92.114 mm Hg

Component A should be the more volatile of the two components (higher P*(T) value) - switch the components if this is not the case.

Bubble Temperature (BUBBLE T) Calculation

Pressure 760 mm Hg

0.5 Benzene 0.5 Toluene

92.116 °C

760.000030516436 mm Hg Bubble T Equation:

542.333 mm Hg

217.667 mm Hg 760.000 mm Hg

0.714

0.286

Dew Temperature (DEW T) Calculation

Pressure 760 mm Hg

0.5 Benzene 0.5 Toluene

98.777 °C

0.9999 Dew T Equation:

0.291

0.709

0 0Estimate the Relative Volatility of the Two Components 1 1

Average Temperature 95.446 °C

Partial Pressure Temperature = 95.446 ˚C Partial Pressure Temperature = 95.446 ˚C

1191.350 mm Hg 483.585 mm Hg

2.46358120977284 0.4059131462901

Equilibrium Data - for y vs. x Operating Line on McCabe-Thiele Diagram

Use Bubble T calculation approach:

0.0 760.0001 110.6 0.0000.1 759.9994 106.1 0.2090.2 760.0006 102.1 0.3760.3 760.0009 98.5 0.5110.4 760.0001 95.1 0.6220.5 760.0000 92.1 0.7140.6 759.9991 89.3 0.7910.7 760.0000 86.8 0.8560.8 760.0000 84.4 0.9110.9 760.0000 82.2 0.9591.0 759.9999 80.1 1.000

z1, FEED z2, FEED

P1*(T) P2*(T)

x1 x2

TBP=

PTOTAL=

P1*

P2* P1*+P2*

y1

y2

y1 y2

TDP=

Sxi

x1

x2

P1*(T) P2*(T)

Relative Volatility aAB Relative Volatility aBA

xA PTOTAL TBP, ˚C yA

P¿=10(A− B

T−C )P¿=10

(A− BT−C )

PTOTAL=x1P1¿(T bp )+x2P2¿(T bp)

∑ x i=y1PTOTAL

P1¿(T dp )+y2PTOTAL

P2¿(T dp )=1

P¿=10(A− B

T−C )P¿=10

(A− BT−C )

α AB=

y A

x A

yBxB

≈

P¿A

P

P¿B

P

=PA

¿

PB¿

COLUMN DESIGN EQUATIONS - TWO COMPONENTS

Parameters for Chosen Set of Components

Component A 9 Benzene Component B 78 Toluene

2.46 1

1 0.406504065

0.5 Benzene 0.5 Toluene

Quality of Feed (q) 0Basis Flow 10 mol/hr

Fenske Equation

0.95 Benzene 0.979269497 Benzene

0.95 Toluene 0.99462004 Toluene

6.5420248922 10.09337884

1.567154753

Underwood Equation

10f 1.7299967254

9.9998936924D 5

15.664310222

10.664310222

2.1328620444

0.95 Benzene 0.979269497 Benzene

0.95 Toluene 0.99462004 Toluene

3.2710124461

Gilliland Correlation

L/D Scaling 1.5 Gilliland Correlation under different Abscissa Ranges:L/D (actual) 3.1992930666 0<x<0.01 -3.716323298854Abscissa 0.2539548932 0.01<x<0.9 0.406433619887186

0.4064336199 0.9<x<1 0.123806185475335

11.70628719

3.2710124461

5.8531435951

aAB aBB

aAA aBA

zA, FEED zB, FEED

Problem A: Given: fractional recovery of A in distillate and B in bottoms

Problem B: Given: mole fraction of A or B in distillate and bottoms product

FRA, DISTILLATE xA, DISTILLATE

FRB, BOTTOM xB, BOTTOM

Nmin Nmin

Nmin, feed

DVFEED

DVFEED, TEST

VMIN

LMIN

(L/D)MIN

Problem A: Given: fractional recovery of A in distillate and B in bottoms - calculate mole fractions

Problem B: Given: mole fraction of A or B in distillate and bottoms product - calculate fractional recoveries

xA, DISTILLATE FRA, DISTILLATE

xB, BOTTOM FRB, BOTTOM

Nmin, feed

[N-Nmin]/(N+1)

[N-Nmin]/(N+1)

Nactual

Nmin, feed

Nfeed, actual

Nmin=

ln { (FRA)DIST (FRB)BOT[1−(FRA)DIST ] [1−(FRB)BOT ] }

ln α ABNmin=

ln {( xA

xB )DIST /(xA

xB )BOT }ln α AB

ΔV FEED=α ABFz Aα AB−φ

+αBB FzBαBB−φ

=F (1−q )

V min=αABDx A ,DIST

αAB−φ+αBBDxB ,DIST

αBB−φ

Lmin=Vmin−D

(FR A )DIST=DxA ,DIST

Fz A(FRB )BOT=

Bx A ,BOT

Fz A

N F=N F ,min( NNmin )

Abscissa=

LD

−(L D)min

LD

+1

N F ,min=

ln {( xAxB )DIST /(zAzB )}

lnα AB

McCABE-THIELE DIAGRAMS FOR SOLVING DISTILLATION PROBLEMS

Parameters for Chosen Set of Components

Component A 9 Benzene Component B 78 Toluene

0.5 Benzene 0.5 Toluene

0.95 0.05

0.05 0.95

L/D T Type "T" for total refluxL/V 1.00

Quality of Feed (q) 0Basis Flow 10 mol/hr

y vs. x Equilibrium Relationship

Use Bubble T calculation approach:

0.0 0.000 0.0000.1 0.209 0.2080.2 0.376 0.3760.3 0.511 0.5120.4 0.622 0.622 A = -0.48110.5 0.714 0.713 B = 1.63160.6 0.791 0.790 C = -2.45940.7 0.856 0.855 D = 2.30810.8 0.911 0.911 E = 0.00040.9 0.959 0.9591.0 1.000 1.000

Feed and Operating Lines

FEED LINE TOP OPERATING LINE BOTTOM OPERATING LINE

Slope 0 Slope 1.00 0.5

Intercept 0.5 Intercept 0.00 0.5

0.05

0.05Slope 1

Intercept 00 0.5 0 0

0.1 0.5 0.1 0.10.2 0.5 0.2 0.20.3 0.5 0.3 0.3

0.399999 0.5 0.4 0.40.4 0.5 0.5 0.5 0 0

0.400001 0.5 0.6 0.6 0.1 0.10.5 0.5 0.7 0.7 0.2 0.20.6 0.5 0.8 0.8 0.3 0.30.7 0.5 0.9 0.9 0.4 0.40.8 0.5 1 1 0.5 0.50.9 0.5 0.6 0.61 0.5 0.7 0.7

0.8 0.80.9 0.91 1

Feed Line/VLE Intercept

If q is not 1: y difference 0.000 If q is equal to 1: 0.500

0.291 0.71348125

0.50023653175

Appropriate Intercept Value: 0.290748918732101

0.500236531746407

zA, FEED zB, FEED

xA,DIST xB,DIST

xA,BOT xB,BOT

The y vs x values are automatically entered into this table if you've solved the bubble T calculation using the Ideal T - 2 cpts worksheet

xA yA yA

Polynomial Fit: Input these coefficients yourself from graph:

x1

y1

x2

y2

xA yA xA yA

xA yA

xA

xA yA

yA

xA

yA

yA=LV

xA+(1− LV ) xA ,DISTyA=

qq−1

xA+11−q

zA ,FEED

yA=LV

xA+(1− LV ) xA ,BOT

PTOTAL=x1P1¿(T bp )+x2P2¿(T bp)

yA=Ax4+Bx3+Cx2+Dx+E

Stepping off Stages

Keep copying the four-row block at the end (ie. A134-->F137) until you have found the total number of equilibrium trays

TRAY NUMBER Variable Value x y

Starting Point: 0.95 0.95 0.95

0.95 0.879 0.9500886

1 0.879 Keep Counting Stages More Equilibrium Trays Required 0.879 0.8794414

0.95008859141 0.742 0.8794815

Step Down 1 0.879 0.742 0.7416845

0.87944141023 0.535 0.7416979

2 0.742 Keep Counting Stages More Equilibrium Trays Required 0.535 0.534965

0.87948152572 0.320 0.5351726

Step Down 2 0.742 0.320 0.3196619

0.74168450125 0.164 0.3200558

3 0.535 Keep Counting Stages More Equilibrium Trays Required 0.164 0.1642626

0.74169789461 0.077 0.1643739

Step Down 3 0.535 0.077 0.0770532

0.53496504934 0.034 0.0760635

4 0.320 Keep Counting Stages More Equilibrium Trays Required 0.034 0.033985

0.53517256136 0.032 0.0711697

Step Down 4 0.320 0.032 0.0317106

0.3196619224 0.014 0.0316374

5 0.164 Keep Counting Stages More Equilibrium Trays Required 0.014 0.013733

0.3200557516 0.012 0.0271551

Step Down 5 0.164 0.012 0.0117375

0.16426260503 0.005 0.011063

6 0.077 Keep Counting Stages More Equilibrium Trays Required 0.005 0.0046427

0.16437393547 0.002 0.0046365

Step Down 6 0.077 0.002 0.0018391

0.07705315819 0.001 0.0018224

7 0.034 Last Required Stage 6.628146953935 Equilibrium Trays Required 0.001 0.0006167

0.07606353373 0.000 0.0005956

Step Down 7 0.034 0.000 8.476E-05

0.03398495891 0.000 6.185E-05

8 0.032 Calculation Complete 6.628146953935 Equilibrium Trays Required 0.000 -0.000146

0.07116966298

Step Down 8 0.032

0.0317105844

9 0.014 Calculation Complete 6.628146953935 Equilibrium Trays Required

0.03163743692

Step Down 9 0.014

0.01373296528

10 0.012 Calculation Complete 6.628146953935 Equilibrium Trays Required

0.02715511304

Step Down 10 0.012

0.01173749506

11 0.005 Calculation Complete 6.628146953935 Equilibrium Trays Required

0.01106303779

Step Down 11 0.005

0.00464273081

12 0.002 Calculation Complete 6.628146953935 Equilibrium Trays Required

0.00463646568

Step Down 12 0.002

0.00183907705

13 0.001 Calculation Complete 6.628146953935 Equilibrium Trays Required

0.00182240738

Step Down 13 0.001

0.00061667271

14 0.000 Calculation Complete 6.628146953935 Equilibrium Trays Required

0.00059560989

Step Down 14 0.000

8.47569661E-05

15 0.000 Calculation Complete 6.628146953935 Equilibrium Trays Required

6.18536326E-05

Step Down 15 0.000

-0.00014648134

xA,DIST

yA

xA

yA

xA

yA

xA

yA

xA

yA

xA

yA

xA

yA

xA

yA

xA

yA

xA

yA

xA

yA

xA

yA

xA

yA

xA

yA

xA

yA

xA

yA

xA

yA

xA

yA

xA

yA

xA

yA

xA

yA

xA

yA

xA

yA

xA

yA

xA

yA

xA

yA

xA

yA

xA

yA

xA

yA

xA

yA

xA

yA

xD line

0.95 0

0.95 1

xB line0.05 0

0.05 1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.0

0.2

0.4

0.6

0.8

1.0f(x) = − 0.48106643378 x⁴ + 1.63156174327 x³ − 2.45938045852 x² + 2.30810850377 x + 0.0004213955

xA

yA

VLE Equilibrium Line

y = x Line

Top Operating Line

Bottom Operating Line

Feed Line

xA,BOT xA,DISTy vs. x VLE Polynomial Fit:

THRE-COMPONENT DISTILLATION: ESTIMATION OF RELATIVE VOLATILITY

Parameters for Chosen Set of Components

LIGHT KEY COMPONENT HEAVY KEY COMPONENT NON-KEY COMPONENTComponent A 78 Toluene Component B 21 Cumene Component C 9 Benzene

Antoine's Equation Coefficients for Toluene Antoine's Equation Coefficients for Cumene Antoine's Equation Coefficients for BenzeneA= 6.95805 A= 6.93619 A= 6.89272B= 1346.773 B= 1460.31 B= 1203.531C= 219.693 C= 207.701 C= 219.888

0.3 Toluene 0.3 Cumene 0.4 Benzene

Partial Pressure Temperature = 50.000 ˚C Partial Pressure Temperature = 50.000 Partial Pressure Temperature = 50.000 ˚C

92.114 mm Hg 18.600 mm Hg 271.236 mm Hg

Bubble Temperature (BUBBLE T) Calculation

Pressure 760 mm Hg

0.3 Toluene 0.3 Cumene 0.4 Benzene

100.306 °C

759.999999724135 mm Hg Bubble T Equation:

168.459 mm Hg

47.006 mm Hg 623.866 mm Hg

408.401

0.270

0.075

0.655

Dew Temperature (DEW T) Calculation

Pressure 760 mm Hg

0.3 Toluene 0.3 Cumene 0.4 Benzene

123.942 °C

1.0000 Dew T Equation:

0.208

0.668

0.123

Estimate the Relative Volatilities of the Two Components

Average Temperature 112.124 °C

Partial Pressure Temperature = 112.124 ˚C Partial Pressure Temperature = 112.124 ˚C Partial Pressure Temperature = 112.124 ˚C

792.987 mm Hg 234.543 mm Hg 1852.492 mm Hg

3.38098783328739 1 7.898303478963

1 0.2957715464559 2.336093434351

z1, FEED z2, FEED z1, FEED

P1*(T) P2*(T) P1*(T)

x1 x2 x2

TBP=

PTOTAL=

P1*

P2* P1*+P2*

P3*

y1

y2

y3

y1 y2 y3

TDP=

Sxi

x1

x2

x3

P1*(T) P2*(T) P2*(T)

Relative Volatility aAB Relative Volatility aBB Relative Volatility aCB

Relative Volatility aAA Relative Volatility aBA Relative Volatility aCB

P¿=10(A− B

T−C )P¿=10

(A− BT−C )

PTOTAL=x1P1¿(T bp )+x2P2¿(T bp)+x3P3¿(T bp )

∑ x i=y1PTOTAL

P1¿(T dp )+y2PTOTAL

P2¿(T dp )+y3PTOTAL

P3¿(T dp)=1

P¿=10(A− B

T−C )P¿=10

(A− BT−C )

P¿=10(A− B

T−C )P¿=10

(A− BT−C )

α AB=

yA

xA

yBxB

≈

P¿A

P

P¿B

P

=PA

¿

PB¿

COLUMN DESIGN EQUATIONS - MULTIPLE COMPONENTS

Parameters for Chosen Set of Components

LIGHT KEY COMPONENT HEAVY KEY COMPONENTComponent A 78 Toluene Component B

3.3809878333

1

0.3 Toluene

Quality of Feed (q) 0Basis Flow 100 mol/hr

Fenske Equation

0.95 Toluene

0.95 Cumene

4.9184225125

0.9992688854

Underwood Equation

100f 0.6010869816

99.999999282D 69.970755417

123.80952017

53.838764748

0.7694466699

0.4073130243 Toluene

0.9490748234 Cumene

0.5712494481 Benzene

2.4171043403

Gilliland Correlation

L/D Scaling 1.25 Gilliland Correlation under different Abscissa Ranges:L/D (actual) 0.9618083374Abscissa 0.0980532419

aAB aBB

aAA aBA

zA, FEED zB, FEED

In a multicomponent system, fractional recoveries are usually specified:

FRA, DISTILLATE

FRB, BOTTOM

Nmin

FRC, DISTILLATE

DVFEED

DVFEED, TEST

VMIN

LMIN

(L/D)MIN

xA, DISTILLATE

xB, BOTTOM

xC, DISTILLATE

Nmin, feed

[N-Nmin]/(N+1)

Nmin=

ln { (FRA)DIST (FRB)BOT[1−(FRA)DIST ] [1−(FRB)BOT ] }

ln α AB

ΔV FEED=α ABFz Aα AB−φ

+αBB FzBαBB−φ

+αCB FzCαCB−φ

=F (1−q )

V min=αABDx A ,DIST

αAB−φ+αBBDxB ,DIST

αBB−φ+αCBDxC, DIST

αCB−φ

Lmin=Vmin−D

(FR A )DIST=DxA ,DIST

Fz A

(FRC )DIST=αCB

Nmin

(FRB)BOT1−(FRB )BOT

+αCB

Nmin

Abscissa=

LD

−(LD)min

LD

+1

0.4878361556

10.555721039

2.4171043403

5.1874923461

[N-Nmin]/(N+1)

[N-Nmin]/(N+1)

Nactual

Nmin, feed

Nfeed, actual

N F=N F ,min( NNmin )

Abscissa=

LD

−(LD)min

LD

+1

Parameters for Chosen Set of Components

HEAVY KEY COMPONENT NON-KEY COMPONENT21 Cumene Component C 9 Benzene

1 7.89830348

0.295771546 2.33609343

0.3 Cumene 0.4 Benzene

Fenske Equation

Underwood Equation

Gilliland Correlation

Gilliland Correlation under different Abscissa Ranges:0<x<0.01 -0.82099578205674

0.01<x<0.9 0.487836155565487

aCB

aCA

zA, FEED

Nmin=

ln { (FRA)DIST (FRB)BOT[1−(FRA)DIST ] [1−(FRB)BOT ] }

ln α AB

ΔV FEED=α ABFz Aα AB−φ

+αBB FzBαBB−φ

+αCB FzCαCB−φ

=F (1−q )

V min=αABDx A ,DIST

αAB−φ+αBBDxB ,DIST

αBB−φ+αCBDxC, DIST

αCB−φ

Lmin=Vmin−D

(FR A )DIST=DxA ,DIST

Fz A(FRB )BOT=

Bx A ,BOT

Fz A

(FRC )DIST=αCB

Nmin

(FRB)BOT1−(FRB )BOT

+αCB

Nmin

N F ,min=

ln {( xAxB )DIST /(zAzB )}

ln α AB

0.9<x<1 0.149678064505704

Parameters for Chosen Set of Components