THE ANALYTICAL CALC"LLATION OF THE NUMBER OF THEORETICAL PLATES

THE ANALYTICAL CALC"LLATION OF THE NUMBER OF THEORETICAL PLATES

By

Gy. S~(RK~(NY, P. R6zSA dud K. TETTA3IANTI

Department of Unit Operations and Department for nIathematics, Technical University, Budapest

(Received October 8, 1969)

I. Historical review

Besides the theoretically exact graphical determination and several approximative solutions, the Lewis' integral [2, 3,4], - taking into account some restrictions enumerated below - is known for the determination of the number of theoretical plates.

I) Equal molal flo'w is assumed in the entire column (i.e. the operation line is linear, R = const ).

2) Concentration change from plate to plate is low enough and constant;

i.e. as shown in a former publication [I] on the general mathematical treat- ment of the problem the operation line and the equilibrium line are linear and nearly parallel. (Linear operation line is supposed implicite in Lewis' calculations as 'well.) The approaching integral of LEWIS [2] derived on the ahove-mentioned conditions is the folIo'wing, using the symbols introduced

in [I]: ^{XD Xr}

(N L)rect = ^I' dx

f(x) - g(x) and

(NdeXh

= J

f(x) -g(x) dx

(I)

Xr x~

where y

=

f(x) is the equilibrium liile function and

y g(x) is the operation line function in the x - y co-ordinate system.

LEWIS made no attempts at the explicit solutions of the integral hut treated a non-ideal mixture, viz. alcohol and water hy using graphical integra- tion.

An explicit solution of the Lewis-integral was found by DODGE and HUFF3IANN [3] introducing a further restriction.

3) The equilibrium curve of nearly ideal mixtures - e.g. that of hydro- x

carbons - can be approximated by the function f(x)

= I-t(x _

l)x where

et;

=

const. The Dodge-Huffmann integral - in the book of ROBINSON and GILLIL...4.ND [4] called the "Lewis Method" - results as follows:

1!l

( 7'; ) _{1 ~ L reet}

=

_{r ?} 2 1 _n [XD+A¹

1!

bi- 4 a1 Cl xT+A1

9 Periodica Polyteehnica Ch, XIYj3-4,

. xT+B¹

j

xD+B1

I I a1 Xb+b1 XD+C1

- n - -

2 a1x}+b1xT+C1 (2a)

322 where

and

and

where

and

CY. S • .fRKAt..-y et al.

a₁

= -

(1X--1)

b^I (x

1)· (1- ~ ⁾ + ~

Cl = _. - - ; XD

R

(x 1) [1

^{..L x ] -IX}

^P

I

R+1+P

^M

R+1+P

(2b)

According to graphical solutions and computations, the value NL ob- tained by the Lewis'-integral, or by the explicit form derived by Dodge and Hllffmann, approximates t1- e "real" value N only if x is sufficiently small.

(The condition 2 for the (onc~ntration change per plate is satisfied for small values of IX only.)

2. A new analytical determination of the number of theoretical plates Condition 2 - being to severe in actual circumstances - restricts the utility of the final integral.

Assuming equal molal flow only a new integral has been obtained for the calculation of the number of theoretical plates in a former paper [1] on

ASAL YTICAL CALCULATION 323

the analytical determination of the problem. The well-known mathematical method, the substitution of optional sections of the y = f(x) equilibrium curve using a polygon of chords is applied (by drawing lines through the smoothed points of measurements) and the partial results obtained for the single chords are summed up further. In the system of equations, constructed from the equations of the chords referring to experimental data and from those of the material balance concerning the single plates; the number of the equations - being equal to the number of plates to be found - is considered unknown and assuming linear operating line - it can be expressed by a recur- sive formula. Using the mean value theorem of the differential calculus, and increasing the number of chords of the substituting polygon, the sums in the expression for the number of equations converge to the following integrals:

Xn ^XI'

- S

f'(x) -g'(x).

- In f'(x)

XI g'(x)

dx ,

S

f'(x) -- g'(x) f(x)-g(x) -j-. In f'(x)

XH g'(x)

dx (3)

f(x)-g(x)

where y = f(x) is the equation of the smoothed equilibrium line and y = g(x) is that of the operating line.

The above restriction for the integral NI, i.e. replacing the curve f(x) by a chord within a single stage - is admittable for every not extremly curved equilibrium line or for relative high numbers of theoretical plates and as a difference to Lewis' integral it ought to be valid for higher values of IX

as well, in consequence of the above mentioned mean-value theorem. For high numbers of the theoretical plates the integral NI is a closer approach- ing to the "real" number N, determinable principally by graphical method only, but showing great technical difficulties for this reason.

0.05 0.5 1.0

I I I

, I

b

0:5

^{x -... ,}

Fig. 1. Graphical determination of the number of theoretical plates on a bevel-angle co-ordinate system

9*

324 ^GY. SARK . .J.'T et al.

Values calculated according to the integral (the method of calculation will be shown later) agree with those of the number of theoretical plates N determined by graphical method assuming equal molal flow. The accuracy of the graphical plotting was entranced by using a large-scale bevel-angle x - y co-ordinate system (cf. Fig. 1). The validity of Lewis' integral for f'(x)

=

g'(x) - i.e. only if the equilibrium line runs parallel with the linear operating line - was proved. As a difference from Lewis' integral the expres- sion for NI (3) can be integrated numerically only, due to the logarithmic function figuring in it. This integral is, however, a generalization of the Lewis one and in the special case f'(x) ...,. g'(x) it tends to the Lewis' integral NL (1).

3. Approximative integration of NI for cc

=

const. and R

=

const The integral NI regarded as the best approximation for the "real"

number of theoretical plates N, can be calculated to any required accuracy, ifthe integration is carried out by a computer. The integral should be expressed, hO'wever, also in an explicit form following the example of LEWIS, or that of DODGE and H ^UFF:MANN. Using an interpolation formula fitting smoothly the measured equilibrium points, the function y

=

f(x) can be approximated.

The adaptation of constant relative volatility x

=

const. is the simplest expe- dient, it is a restriction, ho·wever. On the contrary to LEWIS, the concentration change is not supposed to be the same from plate to plate, the only assumption is the linearity of the function y = f(x) within a stage.

Thus

,x' x

f(x)

= 1+(xD.;'

and

- - ' x , R R+l Applying the approximative relation [5]

In [f'(X) ] ~ _f'(x)=-g'(x) _ = S.

g'(x) [f'(x). g'(x)]O,5 2

the integrand in (3) becomes a rational function which may may be integrated hy elementary functions.

As it was show-n in [5], the geometric mean is a good approximation for the so-called logarithmic mean in a suitable interval:

f'(x[)

i~~x]L

"-' [f'(x)· g'(x)]li²•

In - - g'(x)

A.'·.·lLYTICAL CALCC;LATIOS 325 Substituting into the integral (3), and introducing the denotation

Na =

f

[f'(x)· g'(X)]I12 dx ,

. f(x)- g(x)

we get or

where N H is the "residuum", the error of the calculation, due to the substi- tution, which can be estimated.

After integration

(4i is obtained, where

(4a)

(4b)

The expressions for A and B are the same as the corresponding formulae used by LEWIS, or by DODGE and HUFFl\IANN; the symbols _{a r,} ~r, Cr and ae, be, Ce

respectiyely slightly differ from those of Lewis.

ar

=

(1 x)R; af (1 - x) . (1

f3 ).

R : 1 ' br

=

(x - 1) . (R - ^{XD) ,} be (x

C_r

= -

^XD;

1)-

f3

[1 R+1

. X,vI

1)]

The deriyed approximative integral formula (N_a) is similar to that of LEWIS (N

d

but being simpler, the integration is less laborious.

Numerical calculations showed, that from 20 plates upwards Eq. (2) and Eq. (4) approximating the more exact expression Nj, give a slightly differ- ent result from that of the graphical construction made stepwise for N and the difference between Na and NI can be estimated. The quoted paper [5]

contains the estimation of the error in Na referring to NI; (NR = Na - NI).

NI - being a more general term than .N L - gives more exact results than that of Lewis in spite of its approximative character. It should he noted, however, that Lewis' criteria for the concentration change, being constant from plate to plate for small values off - g, i.e. at high number of theoretical

326 Cl". SARK.·it .. .,. et 01.

plates, judged too severe at first, do not seem to be important in the light of numerical results. The reality of Lewis' restrictions is demonstrated by it.

For IX

>

2, the error of the Lewis' integral becomes significant; the error

of the approximative term Na becomes negligible as long as the error N R

caused by the above mentioned substitution where remains sufficiently small.

Z= j'(x)-g'(x) g'(x)

4. Determination of the value Nmin

FENSKE'S equation [7] is known for the analytical determination of N_min (R =

=,

or g = x, g' = 1):

(5)

The equation of FENSKE is of general validity, even the theorem of equal molal flow is disregarded in it, thus the value of the calculated Nmin

renders the value of N_min obtained graphically.

In practical application, however, the change of the values of IX11 from plate to plate causes difficulties, therefore the equation is applied for ^7.

=

const.

only:

IV_min =

In xD(1 XM) XM(l - XD)

In IX (5a)

A similar equation to that of FENSKE can be derived from the integral NL of LEWIS (cf. Eq. 2) as well [4].

lim N ^L

=

(N ^dmin

=

~ In --=:-'---"':';';:'=-'

R.-,.oo 7.-1 X,\1(1-XD)

(6)

IX must be equal or nearly equal to unity to get the equation proposed by FENSKE.

The integral 1\[] will be integrable taking into account the restriction

(IX - l)x ~ 1; further g = x and g' = 1 respectively; thus the equation of Fenske is obtained in the following form:

-X.vf)

In - -

xM(I--xD)

In ^0: (7)

ANALYTICAL CALCULATION 327

Considering the approximative integral Na. the adequate form of the equation of FENSKE is as follows:

If- I xD(I-xM) to:;' n -=---'---=-'- x!vI(I-xD)

0:;-1 (8)

x - I Comparing Eqs (7) and (8), it is easy to observe: the expression IS III

1'0:;

(Na)min instead of the adequate expression In 0:; in N min; according to the approximative relation [5]:

IX 1 ,_

- - "-'Vx,

lnx

Thus, (Na)min can be used instead of the expression N min as long as the loga- rithmic mean can he approximated by the geometric mean with satisfying accuracy.

5. Gilliland's universal graph for the determination of the number of theoretical plates

The well-known form of GILLILAND'S generalized diagram is as follows:

,( R - R min )

q; R

+

^{1 (9)}

where R denotes t1e reflux.

According to the opinion of the author: " ... there should he no unique correlation of this form ... " [6] the approximative integral Ncr and the formula of Lewis N ^L as well shows that the universal character of Gilliland's diagram cannot he supported theoretically; i.e. neither the function N L(R) , nor the function Na(R) can he expressed in the form (9).

Our attempts didn't even succeed in finding the reason, why GILLILAND'S data are clustered in a relative narrow hand. GILLILAND'S diagram is to he applied carefully for the extreme values of 0:; and

p.

The verification of the universality of Gilliland's diagram was tried using numerical examples calculated with the approximative integrals Na•

The results can he summed up as follows:

a) for x

=

const. (1.18 - 2.5) and R = const. (1.03 - 23)

328 GY. SARKAiYY et 01.

R Xl'

~----~--._.

I I I

I

1.18 I _: 23.03 0.25 0.85 0.07

I 0.15

i 1

- 2.5

2

2

I

⁵

; 1

I

I

I

1 3.5

I i

!

I

1.03

N-Nmin

~ 1.0.

0.5

I

I 0.5

0.5

0..9

W'

^{I , , '" , , , ,}

0..8 I I I !

'~ rH-i+ " ,

t(

^{' ,}

_,

^{" ' 1 ,}

_I

^I

1 ' , 11 I

0..7 0..6

1111 t

I , , , , ,

" , , , , I , , , , 11 I

0..5

I I 1 1 I

! 0.85 0.001

, i 0.95

I _0.05

i ,

0.25

i

0.95 0.05

,

i 0.4

1---

0.66 0.05

0.75

0.66 0.25

0.35

I

I I

I I

0..4 ~~'rtJ ⁴²

I I

~~2! ^{I I ,} ' ,

I

ⁱ

I , ,

" , , , I i

i

1 , _{, ,} , _{' '31}

I

32 ^{1 I}

I

I

11 1 , 1

11 I t , , _i

H+1 ~ 1»j-1"H::1 I

I I I ^{I 44rl,.3}

~2_'_"_" I t i l l ' I

; 2~*'H-f-j ^{23 , '21}

I I ⁱ I I

I !

0..3 0..2

0..1 0.

3.33 6.0 0.7 0.902 1.0 1.8 1 4.5 0.37 0.55 0.66 1.11

I

I I I I ^I

! i

!

_,

"'-...

0. 0..1 0..2 0..3 0..4 0..5 0..6 0..7 0.8 0.9 1.0.

R-Rmin

R+1

Fig. 2. Gilliland's curve

No.

_._-- -

I I 11

i

¹²

- 21 22 23 24 31 32

-

41 42 43 44

No.

11 12 21 22 23 24 25

- - I

31 ⁱ 32 I

I I

A:>AL YTICAL CALCL"LATIO,y 329

XM _1 __ ^R:.

P

-.----~". - - - ^..-

1

3.7

~--..

i ^{1.05 3.0}

}

^{0.25 0.99}

O~~_J

- 1.0

I I

I I

0 ^0.66

i

I

!

_,

I

N-Nmin

~

1.0 i

0.9

I

~

^{1 1 1 I} I ( . I , 1

" ,

•

^{I1 j i}

^I

, I i I ,

I11 I I

11 I I

0.8

=

^{• I ~~i1* 1 1 : I 1 ' 1 I 1 \ I 1 1 i ! 1 i}

0.6

' 1 , 1 i

! I 1"Ii 111221

, , I'\.)' I..:.., I

2.5 1.18 1.5 2 2.5 1.18 2.0

0.5

I

~d31tH

I ' 1 1 : I ' ~: :+:2t~

l I

, I

I

₁

0..4

'-

¹¹¹¹¹¹

H-ft.~

0.5 0.95 0.05

0.46 0.72 0.07

0.50 0.66 0.25

I I

I

,

I

I

1

I J

I I

1 I

I

I

W

I

I ^{\ I}

h

1

I

11 I I l l ' l ' , 24

I I

¹

0.3 0.2 0.1

0.

1

^{I 11 , , , !-....: , 1} ' 1 1 : I I ^{I I i}

^{I I}

^!

1 ' , , , ... , , ' I 11

1

1 1 . i I 1 -., ' ,

i i ' ! I I : , " ,

I

i

I

I

[32 : ~ :--,c I I I : I I ; : I

I

¹ , : : : : : : , I I I I I I 1 ' 1 ?~ ~ ^{I 11} ~ ^{i : , 1 I} ,

I ^!

^I

i

I

I ^~

i I ^{' " I}

I 1 I I i±i±ti

.---...J

o

0.1 0.2 0.3 0,4 0.5 0.,6 0..7 0.8 o.,g la

Fig. 3. Gilliland's diagram

R-Rmin

~

I

variable 1.2 1.21 1.43 2.06 4.55 1.3

co

only for strippng

330

N-Nmin

N+1

Cl'. S . .fRKAI'il' el al.

1.0 ~-,--,--'----.---r--'---'---""--'----'

0.9 0.8

5

0.7 ~-m.:l+H---+---+---+--+--+-""\--""\--""\---1 0.6

0.5 0.4 0.3 0.2 0.1

o

I I

I I , I

I ,

, i I , I I I , I :

:

^I

I I I I

i

I I I , I I

i

I , I I

I _ - I I ,

i

I I I I I I I I

I I

o

0.2 03 0.4 0.5 0.6 0.7 0.5 0.9 1.0

Fig. 4. Gilliland's diagram

l.fJ=17,8

1

? fJ- 84

3:

^{fJ: 3:7 1 O·}

4. fJ = 0,88

J

^{et: = .;,}

5. fJ = 0,3428 6. fJ 0,175

R-Rmin

R+1

the values of the function in (9) are varying in a large interval d·epending on the parameter

f3

(0.37 - 6.0) (cf. Fig. 2)

b) for

f3

= const. and l.05

<

^x

<

³

the values of the function in (9) are varying in a relative narrow band being close or less close to the GILLILAND'S diagram depending on

f3

(cf. Fig. 3)

c) for x = l.05

XD = 0.99 Xi"r = 0.05

large deviations (50-100 per cent) can be found depending on extremly scattered values of XT and f3 respectively (f3 = 0.175 -17.8) (cf. Fig. 4).

ANALYTICAL CALCULATION 331

It is to be emphasized, that these significant deviations from Gilliland's curve are due to the extreme values supposed here to act, and which were chosen ,~ith intent so as to eschew "inner compensation" and so as to give a bias to the curve.

Though Gilliland himself did mention that in the construction of his diagram "extreme" conditions 'were considered as much as possible, it seems likely that, in most of the cases, the narrow band was due to inner compen- sations.

Since, deservedly, Gilliland's diagram is widely popular in instructional circles, we thought it not amiss to suggest these critical notes on it.

Summary

The formulae proposed by LEWIS, and DODGE and HUFF1IAl'il'i, respectively impose rather strict restrictions on the determination of the number of theoretical plates of columns for the rectification of binary mixtures. These relations are referring to relative low volatility (0:) or high number of theoretical plates (N) only.

The results, obtained by our method, are summed up as follows:

1. A generalization of the integral of LmYIs is proposed.

2. Constant molecular flow is the only condition imposed, obtaining the integral valid for each equilibrium curve y = f(x), and for each N respectively.

3. Analytical approximation Na to integral NI is proposed for 0: = const. The error of the approximative formula can be estimated.

The universal diagram of GILLILAl'iD was controlled by calculations using the expres- sion ?'i_a•

Symbols N 'real' number of theoretical plates

lYL Lewis' approximative integral for the number of theoretical plates (Eq. (1».

integrated by Dodge and HlIffmann (Eq. (2»

NI Authors' approximative integral (Eq. (3»

lV_a approximative expression of the approximative integral NI (Eq. (4.»

NR Nu -N,: The error in Na referred to !'v_J R Reflux ratio

Subscripts:

- D Refers to distillate

- M Refers to bottom

IT!

Refers to feed Index of NR

Refers to mean value of Cl:

Refers to rectifying column

Refers to exhausting (stripping) column References

1. JUl'iG, G.-R6zs.-I-, P.-S.~RK.~NY, Gy: Pub!. :Mat. Inst. Hungarian Acad., Sci,. I!., 227 (1957)

2. LEWIS, W. K.: Ind. Eng. Chem. 14, 492 (1922)

3. DODGE, B. F.-HuFBIANl'i, J. R.: Ind. Eng. Chem. 29, 1434 (1937)

4. ROBIl'isol'i, C. S.-GILLILAND, E. R.: Elements of Fractional Distillation, 4th ed., 1950.

5. TETTA~IANTI, K.: to be published in this Journal 6. GILLILAl'iD, E. R.: Ind. Eng. Chem. 32, 1220 (1940) 7. FENsKE, M. R.: Ind. Eng. Chem. 24, 482 (1932) Prof. Dr. Karoly TETTAMAl'\TI]

Prof. Dr. Pal R6zSA Dr. Gyorgy S_tRK_-\'Z"Y

Budapest XL, I\Hiegyetem rkp. 3, Hungary