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DISTILLATION SEPARATION FACTOR

STUDIES FOR IDEAL AND NON-IDEAL

SYSTEMS

Zawar Ahmad Khan

Supervisor

Dr. K. M. Bukhari

Co-Supervisor

Dr. Muhammad Aslam

Center for Nuclear Studies

Quaid-i-Azam University, Islamabad, Pakistan

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OBJECTIVE

To study the Distillation Separation Factor.

For theoretical study H2O-HDO-D2O system (ideal system) was selected, becauseexperimental study of this system requires sophisticated and costly equipment, like

mass spectrometer etc.

For experimental study H2O-C2H5OH system (non-ideal system) was selected as

theoretical study of this system is quite complicated.

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Theoretical Analysis

3.1 DERIVATION OF RELATIONSHIP (K = Ae-B/T)

This relationship is derived for exchange reaction H2O + D2O 2HDO in vapor phase .

For a typical reaction: A + B C + D (3.1)

the equilibrium constant K can be determined from the expression

K = [C] . [D] / [A]. [B] (3.2)

and RT lnK = -G (3.3)

In the case of isotopic reaction, the G quantities are very small and it is not possible to calculate them from Eqn. (3.3),

since the uncertainties with which the molar free energies of the various isotopic species are known, are generally greater

than G. Furthermore, in statistical thermodynamics one can demonstrate that, for one mole of perfect gas

G = -RT ln(Z/N) (3.9)

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Where Z is the partition function for the system under consideration.

Comparing Eqn. (3.9) with Eqn. (3.3) we have

K = ZC.ZD/ Z

A.Z

B (3.10)

Statistical mechanics thus permits the calculation of equilibrium constants

through the partition function Z, relevant to the reaction of various molecular

species. Since the total energy for each level can be broken down into theterms relating to the different degrees of freedom, we find that

Z = Ztrans. Zrot. Zvibr. Zel. Znucl. e-Eo/kT (3.11)

In which the first five factors refer in order to translational, rotational,

vibrational, electronic and nuclear degrees of freedom, and the final termexpresses the zero-point energy Eo.

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For isotopic exchange reactions of the general type:

aAXb + bB*Xa aA*Xb + bBXa (3.12)

In which X and *X are two isotopes of the same element and a and b are the number of moles

of different species

The equilibrium constant is

K = (*Z/Z)a AXb/ (*Z/Z)b

BXa (3.13)

Introducing the notation u = h/kT we can write the *Z/Z ratio that appears in Eqn. (3.13) as

*Z/Z = (*Z/Z)trans.(*Z/Z)rot.(*Z/Z)vibre()(*u-u) (3.15)

Where Ztrans= const(M3/2. T5/2)/P (3.16)

And Zrot= const. IT/s (3.17)For diatomic or linear polyatomic molecules (where M is the molecular weight, P is the

pressure, I is the moment of inertia and `s` is the molecule's symmetry number), or

Zrot= const.(Ix Iy Iz)1/2.T/s (3.18)

For non linear polyatomic molecules Ix, Iy and Izare the principal moments of inertia. The

partition function for the vibrational degree of freedom is given by :

Zvibr= [1/(1-e-u)] (3.19)

In which the product is extended to all the fundamental frequencies of the molecule. For

diatomic molecules, with only a single fundamental frequency, we obtain from Eqns. (3.16),

(3.17) and (3.19).

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*Z/Z = s/*s(*M/M)3/2.(*I/I).[e-*u/2/(1-e-*u) ].[(1-e-u)/e-u/2] (3.20)

Where as for nonlinear polyatomic molecules, we obtain from Eqns. (3.16),(3.18) and (3.19)

*Z/Z = [(s/*s)(*M/M)3/2.(*Ix*Iy*Iz /IxIyIz)1/2]{[e-*u/2/(1-e-*u)].[(1-e-u)/e-u/2]} (3.21)For polyatomic molecules the difficulties can be overcome in many cases bymaking use of the rule of the product of frequencies, stated by O. Redlich[86], which gives

(*M/M)3/2.(*Ix*Iy*Iz /IxIyIz)1/2= (*u/u) (3.22)

Eqn. (3.21) then becomes

*Z/Z = (s/*s){(*u/u).[e-*u/2/(1-e-*u)].[(1-e-u)/e-u/2]} (3.23)This equation is also applicable to diatomic molecules, yielding results veryclose to those obtained with Eqn.(3.21). Eqn. (3.23), developed in powersseries of of u = *u-u and approximated except for terms of a higher orderthan the first, becomes:

*Z/Z = (s/*s)eG(u).u (3.24)

With G(u) = 1/2 - (1/u) + 1/(eu-1).

Tabulation of the G(u) function [35], which varies from 0 to 1/2 for uvariable between 0 and , makes it far simpler to calculate the equilibriumconstants [87].

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APPLICATION TO EXCHANGE REACTION

H2O + D

2O 2HDO

H2O + D2O 2HDO or

OHH + ODD OHD + ODH (3.25)

Let a = b = 1; A= OH, B = OD, X = H and *X = D Using Eqs.3.12 and 3.25,

Eqn 3.13 becomesKvap=(*Z/Z) / (*Z/Z) HDO = (*Z) . (3.26)

Kvap= (Z)2

HDO/{(Z) .(Z) } (3.28)

Specifying the quantities of Eqn. (3.28) we have

Kvap=(Z)HDO.(Z)HDO /{(Z) .(Z) .}= ( . /sHDO.sHDO)

(uHDO.uHDO/ u .u ){exp[-(uHDO+ uHDO- - )]}f(u) (3.29)

where f(u) = [1-exp(- )] [1-exp(- )]/[1-exp(-uHDO)]2

or neglecting the factor f(u) which at 1000 K differ from unity by less than 4%

we have:

OH2

OH2 HDOZ)( }).(*)/{( 2 HDOOH ZZ

OH2 OD2

OH2 OD2 ODOH SS 22 .

OH2 OD2OH2u OD

2u

OD2u OH2u

Kvap= ( . / s2

HDO )(u2HDO/ . ).{exp[-(2uHDO- )]} (3.30)OH2S OD2S. OH2u OD2u

ODOH 22 uu

0

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Sample Calculations

Consider the exchange reaction:

(H2O)vap+ (D2O)vap (2HDO)vapFor the calculation of its equilibrium constant Table 3.1 below showsthe numerical values of the quantities required for the calculations.These calculations are made by neglecting anharmonocity correctionsand by using Eqn. (3.30).

Table 3.1. Molecular constants of H2O, D2O and HDO [20].Constant H2O D2O HDO

M(a.m.u) 18 20 19

(o/c)(cm-1) 4671 3430 4077

uT (K) 6785 4938 5871

s 2 2 1

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Plugging in the values of s's and u's in Eqn. (3.30) from Table 3.1, we haveKvap= [2x2/(1)

2]x[(5871)2]/[6785x4938)]x{exp[-(1/2T)(2x5871-6785-4938)]}

Kvap= 4.115 exp(-10/T)

Kvap(at 273 K) = 3.967

Kvap(at 298 K) = 3.979

Kvap(at 348 K) = 3.998 Relationship for Kvapis also determined by using Eqn. (3.23) and using the three

fundamental frequencies for every three isotopic molecules of water and the

relationship found is:

Kvap= 4 (0.753) (1..043) (1.3248) Exp (-19/T)

Kvap= 4.1732 Exp (-19/T)Kvap(at 273 K) = 3.893

Kvap(at 298 K) = 3.915

Kvap(at 348 K) = 3.951

The corresponding error found for the three temperature are 2.0%, 1.53%, and 1.27%.

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Table 3.25. Review of liquid and vapor phase equilibriumconstant data for the exchange reaction H

2O + D

2O

2HDO as a function of temperature by different workers.

Referenc

es

Temperat

ure C

Urey

1947

(data of

Libby and

Herzberg

1943)

K(HM)

[28]

Kirshenb

aum

1951

[35]

Bigelesi

n 1955

(Theo.)

[36]*

Pyper R. S

Newbury

and G. W.

Barton Jr

(1967) Kl

(Exptl.)

[37] A. J.

Kresgi

Chiang

(Substrate

Method)

1968 Kl

(Exptl.)

[38] J.

R.

Holston

(Calc.)

1969

[39] Wayne

C. Duer

and Gray L.

Bertand

(1970) K

(Exptl.)

[41] J. W.

Pyper, R.

J.Dupzyk, R. D.

Friesen, S. L.

Bernasek et.el

(1977) Kv

This

wor

k

199

8

0 3. 94 3.76 - - - 3.69 3.74 0.07 - 3.967

25 3.96 3.80 3.96 3.75 0.07 3.85 0.03 3.72 3.76 0.02 3.81 0.09 3.97

9

75 3.98 3.85 - - - 3.78 - - 3.99

8

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ConclusionsTheoretical values of gas- phase equilibrium constant are calculated for

different spectroscopic data reported by different authors, from 1933 to 1997

using Eqn.(3.30).

1. Generally theoretical calculations of liquid-phase equilibrium constant are

quite difficult due to the involvement of intermolecular forces. But this is not

true in case of isotopic species as intermolecular forces are isotope

independent and cancel out in ratios. Also the required data is not frequently

available.

2. Harmonic data gives values more close to the classical value of 4.0 i.e. as

high as 3.98.

3. Anharmonic data gives values quite smaller than the classical value of 4.0 i.e.

as low as 3.20.

4. Semi empirical values of Kvand Klshow that at low temperatures i.e., up to

298 oK, Kv Kl, but at higher temperatures KlKv .

5. Values of K increases with the increase in temperature for this exchange

reaction.

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SEPARATION FACTOR FOR H2O-HDO-D2O

SYSTEM

A theoretical approach is applied to develop a relationship between

the separation factor (

) for nearly ideal System i.e., H2O - HDO - D2O,

and the equilibrium constant for the exchange reaction H2O + D2OHDO

in liquid and vapor phase.

The resulting equation is of the type (distillation) = f(Kl,Kv) where Kland

Kv are equilibrium constants in liquid and vapor, where K = Ae-B/T

DEVELOPMENT OF SEMIEMPIRICAL RELATIONSHIP FOR

DISTILLATION SEPARATION FACTOR OF H2O-HDO-D2O SYSTEM

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DERIVATION OF RELATIONSHIP

[ (distillation) = f(Kl, Kv)]

Considering the exchange reaction in liquid phase:

H2O (liq) + D2O (liq) 2HDO (liq) ( 4.18)

The equilibrium constant for the reaction (4.18) is

Kl= [HDO]2liq /[H2O]liq [D2O]liq ( 4.19)

Applying Daltons Law of partial pressure and Roaults Law of ideal solution of mixtures

We arrive at Kl/ Kv= (4.23)

As HDO cannot exist in the absence of H2O and D2O its vapor pressure cannot be determined

directly and we have to make certain assumptions.

Assumption-Iis the geometric mean of and i.e.,

(4.24)

Therefore Eqn. (4.23) gives Kl/ Kv= 1 or Kl= Kv (4.25)

For separation to be possible KlKv, therefore this assumption is not valid [91].

2000 )/(22 HDOODOH PPP

0

2OHP

0

2ODP

)( 000 22 ODOHHDO PPP

0

HDOP

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Assumption-II

is the arithmetic mean of and i.e.,

= ( + )/2 (4.26)

Subsituting Eqn. (4.26) in Eqn. (4.23) we get

Kl/ K

v= 4 /( + )2 (4.27)

Kl/ Kv= ( / ) + ( / ) +

But / 2, from Eqn. (2.16); is used as it is assumed that GM rule is notexactly obyed.

Therefore Kv/ Kl(2/4) + (1/42) + (1/2) (4.28)

The operational range of is 1.07 at 298K and 1.02 at 373K.

Let = 1.07 then Eqn. (4.28) gives Kv/ Kl1.0046Therefore Kv1.0046Kl, i.e., Kv> Kl

Also for = 1.02; Kv1.0004 Klor Kv> Kl

This result is contradicting the experimental data in this temperature range

[28]. Hence this possibility is also not valid.

0

HDOP

0

2OHP 0

2ODP

0

HDOP

0

2OHP

0

2ODP

0

2OHP

0

2ODP

0

2ODP

0

2OH

P0

2OH

P0

2ODP

0

2ODP

0

2OHP

0

2OHP

0

2ODP

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We can assume: (i) when is slightly greater than 1 then log = (

-1) and

(ii) (+1)/2 1

Therefore Eqn. (4.31) becomes

Exp(

(Kl/ Kv- 1))(4.32)

If C is the correction factor, then above Eqn. (4.32) becomes

= C Exp((Kl/ Kv- 1))

(4.33)

As decreases and the ratio Kl / K

v increases with the increase in

temperature as shown in Table 4.16 the negative sign of exponentialpower will lead to correct solution.

EXPANSION OF RELATIONSHIP

Taking log of both sides of above Eqn. and approximating:

Exp{(Bv-B

l) /T} = 1+(B

v- B

l)/T, we get:

log = log Cavg- [ (Al/Av) { 1 + (Bv- Bl)/T }- 1 ]1/2

(4.34)

If K A e , K A e and C C , then Eqn. (4.33) becomes:

= C e

l l

B /T

v v

B /T

avg

avg

-A

Ae 1

l v

l

v

Bv Bl

T

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CALCULATION OF CORRECTION

FACTOR (C)

Firstly (Kl, Kv) data quoted in [30] is used for the

calculation of factor

Exp((Kl / Kv - 1) ).

Secondly is calculated using empirical Eqn. quotedin [63 ] i.e;

log = log ( / ) = (019645.4/T ) -(48.5026/T) +

0.013615

Then expression = C Exp((Kl / Kv - 1) ) is used for

the calculation of correction factor C.

Values of these factors are reported in Table 4.16.

PH O2

PHDO

21

21

2

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Temp. K Kl[28] Kv[28] Kl / KvExp (-(Kl / Kv- 1)

) C

270 3.42 3.448 0.99188 - - -

280 3.46 3.475 0.99570 - - -

290 3.50 3.497 1.00086 0.9711 1.083 1.11523

300 3.53 3.517 1.00370 0.9410 1.073 1.14030

310 3.55 3.533 1.00481 0.9331 1.064 1.14029

320 3.57 3.551 1.00540 0.9292 1.0554 1.13582

330 3.59 3.568 1.00620 0.9243 1.048 1.13383

340 3.61 3.583 1.00754 0.9170 1.042 1.13650

350 3.63 3.598 1.00890 0.9100 1.036 1.13864

360 3.64 3.621 1.00780 0.9155 1.031 1.12620

370 3.66 3.625 1.00970 0.9062 1.026 1.13220

380 3.67 3.638 1.00880 0.9105 1.0223 1.12280

Table 4.16. Calculated values of C and data used in its calculations

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Table 4.17 Data for the development of empirical equations and

evaluation of constants Al, Av, and Bl, Bv.

Temp.

K

X-axis

(1000/T)

Y-axis

(ln Kl)

Y-axis

(ln Kv)

270 3.704 1.22964 1.2378

280 3.571 1.24130 1.2456

290 3.448 1.25280 1.2519

300 3.333 1.26130 1.2576

310 3.226 1.26700 1.2622

320 3.125 1.27260 1.2672

330 3.030 1.27820 1.2720

340 2.941 1.28370 1.2762

350 2.857 1.28920 1.2804

360 2.778 1.29200 1.2843

370 2.703 1.29750 1.2879

380 2.632 1.30020 1.2914

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RESULTS

Two curves are are plotted in Figures 4.1 and 4.2 using data given in

Table 4.17. Empirical equations and values of constants evaluated are

given below

4.2.4.1 Statistics and fitted empirical equation for Kl(i) Statistics:

log Kl= -57.91/T + 1.4536 (4.35)

Regression sum of squares = 0.00226535

Residual sum of squares = 4.77678 E-006

Co-efficient of determination R2= 0.997896

Residual mean squares = 5.97097 E-007(ii) Empirical equation developed is of the type:

Kl= 4.2785 e-57.91/ T

4.2.1.1 Statistics and fitted empirical equation for Kv

(i) Statistics:

log Kv= - 48.463/ T + 1.41885 (4.37)

Regression sum of squares = 0.00158652Residual sum of squares = 2.6971 E-007

Co-efficient of determination R2= 0.99983

Residual mean squares = 3.3713-008

(ii) Empirical equation developed is of the type:

Kv= 4.132 e-48.463/T

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SAMPLE CALCULATIONSEmpirical equations obtained are given below:

Kl= 4.2785 e- 57.91/T

and Kv= 4.132 e- 48.463/T

The values of empirical constants obtained from these equations and

Table 4.16 are listed below:

log Cavg= 0.124155; Al= 4.2785, Bl= 57.91; Av= 4.132, Bv= 48.463

Plugging these values in Eqn. (4.34); we get

log = 0.124155 - [0.0355 - (9.782/T)]1/2 (4.39)

This equation is valid for the temperature range of 283 K to 370 K.

(i) Now at 300 K, log = 0.124155 - 0.0538 = 0.0704,

therefore = 1.073

(ii) And at 373 K, log = 0.124155 - 0.096306 = 0.02785,therefore = 1.028

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COMPARISON OF RESULTS

Table 4.18. Comparison of results of separation factor for H2O-HDO-D2O system of various workers as a function of temperature.

Re

f.N

o.

Temp. K 273 283 293 298 303 313 323 333 343 348 353 363 373

[6

3]

Lewis and

McDonald

1933(Calc;)

1.079 1.069

3

1.060

8

1.05

33

1.04

7

1.0406 1.0353 1.031 1.02

67

[2

1]

B. Topely

H. Eyring

1933

(Theoretical)

1.092 1.083 1.07

1

1.067 1.060 1.05

3

1.03

8

1.02

5

[5

6]

Riesenfeld

and Chang

1936 (Calc;)

1.082

2

1.071

4

1.062

9

1.056 1.04

94

1.04

37

1.0386 1.035 1.0298 1.02

58

[5

5]

Miles and

Menzies

1936

(Empirical)

1.093

6

1.079

5

1.069

8

1.061

3

1.05

34

1.04

65

1.0404 1.0348 1.030 1.02

58

[2

8]

Kirshenbau

m

1951

(Calculated)

1.12 1.087 1.07 1.066 1.059 1.05

2

1.02

6

[6

1]

Combs

Googin and

Smith 1954

(Calculated)

1.093

6

1.082

2

1.071

4

1.063 1.05

45

[6

1]

Combs

Googin and

Smith 1954

(Exptl.)

1.100

3

1.087

3

1.08

2

1.075 1.063 1.05

1

[5

9]

E. Whalley

1957

(Empirical)

1.109 1.07

2

1.05

0

1.03

5

1.02

5

[6

4]

W. M. Jones

1968(Calc.)

1.088

2

1.078 1.068

2

1.060 1.05

3

1.04

62

1.0403 1.0351 1.0304 1.02

61

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Table 4.18 continued........

Ref.

No.

Temp. K 273 283 293 298 303 313 323 333 343 348 353 363 373

[65] W. Alexander

Van Hook

1968(Empirical)

1.105 1.075 1.054 1.038 1.028

[66] Jovan Pupezine

et. al 1972

(Empirical)

1.1076 1.0932 1.081 1.0703 1.0612 1.0533 1.0464 1.0404 1.0352 1.0306 1.0267

[66] From salt

solutiondata (+0.3)

(Empirical)

1.1074 1.093 1.0807 1.0701 1.061 1.0531 1.0462 1.0403 1.0352 1.031 1.027

[38] J. H. Rolston

1976 (Empirical)

1.113 1.098 1.080 1.074 1.065 1.056 1.040 1.027

[69] Gy Jakli and

G. Jancso` 1980

(Empirical)

1.107 1.093 1.076 1.071 1.062 1.054 1.038

[72] Rong Sen Zhang

1988 (Empirical)

1.108 1.0932 1.081 1.075 1.070 1.061 1.053 1.0463 1.0404 1.038 1.0353 1.031 1.027

This work 1998

(S. Empirical)

Eqn. (4.39).

1.098 1.0813 1.0751 1.070 1.061 1.0533 1.047 1.0414 1.039 1.03654 1.032 1.0282

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Discussion

Previously it was thought that H2O-HDO-D2O system is a perfectlyideal mixture, and it was also assumed that vapor pressure of HDO is

the geometric mean (GM) of the vapor pressure of H2O and D2O, asHDO exists only in the presence of H2O and D2O, and cannot beisolated [21, 52, 53, 35]. But later on several workers have reportedthat this system deviates from perfect ideality and GM rule is notexactly valid [30, 65, 66, 67, 69, 90]. The main problem in water is to

be able to distinguish between the contributions arising from deviation

of the(i) GM rule for the vapor pressure of HDO (ii) that arising from thenon-ideality of liquid mixture (iii) and non ideality of the vapor, in thesense of deviation from perfect gas equation. We have attributed the alleffect to non-validity of GM rule.

Our semi-empirical model is based on the assumption that vaporpressure of HDO is the harmonic mean (HM) of vapor pressure of H2Oand D2O, and a comparison of our results with those of other workersis given in Table 4.18 and Figure 4.3.

122

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273.00 293.00 313.00 333.00 353.00 373.00283.00 303.00 323.00 343.00 363.00

Temperature [K]

1.02

1.04

1.06

1.09

1.11

1.03

1.05

1.08

1.10

1.12

Separa

tionFactor

Figure 4.3. Separation Factor vs Temperature curves representing data of differentworkers.

Separation Factorvs Temperature Plotsof different Workers

Ref. 61

Ref. 19

Ref. 54

Ref. 53

Ref. 26

Ref. 59[Calc;]

Ref. 59[Exptl;]

Ref. 57

Ref. 62

Ref. 63

Ref. 64[Empirical]

Ref. 64[salt solu; data]

Ref. 36Ref. 67

Ref. 70

This work 1998

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In[63] it is stated by Lewis and R. T. McDonald that [H(HDO)]liq[H(HDO)]vapdue to intermolecular forces in liquid phase.

It is also well known that [S(HDO)]liq[S(HDO)]vap, and as

G = H - S. Therefore [G(HDO)]liq[G(HDO)]vap.

Also it is known fact that

- G = RT ln K or K = Exp (- G/RT ) from where it is evident that KvKl, which proves that HM rule is applicable instead of GM or AMrule for

In [30] it is proved by Alfred Narten that geometric mean rule is not

obyed.

In [65] the non-ideality of H2O-HDO-D2O system is proved by

Alexander Van Hook.

In [67] the deviation from geometric mean rule is shown to be from0.02 to 0.08 by Alexander Van Hook.

PHDO

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In [68] and [69] Gy. J. Akli et. al also show the deviation from ideality

of H2O - HDO - D2O system and we have proved the validity of HM

assumption from their data and

KlKvindicates that separation is possible.

Kl- Kv) decreases with increase in temperature which explains the fact

that decreases with increase in temperature. Values of calculated from our semiempirical model match well with

the theoretical and empirical values of many other workers. The

values found by us are also in good agreement with the experimental

values quoted in litrature and it is safer to use our relationship in the

temperature range 295 K to 345 K.

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EXPERIMENTAL STUDY OF H2

O-C2

H5

OH

SYSTEM

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Experiments were performed to determine the rates of evaporation in ethanol-watermixtures under different conditions. The objectives of these experiments are to studythe separation of water from ethanol by using evaporation with forced circulation of airat ambient temperature.

APPARATUS

The major apparatus used along with their functions are given below :

i) Solution density measurement apparatus DMA-48 of Anton Parr was used todetermine the Vol% Ethanol in Ethanol-Water mixture.

ii) Air Drying unit (AD) which contains CaCl2. Air is sucked through this unit tomake it as dry as possible.

iii) Air flow rotameter (AFM) to measure the flow rate of air (range 0-3000 lt/hr).iv) Liquid flow rotameter (LFM) to measure flow rate of liquid feed mixture

(range 0-60 lt/hr).

v) Thermometer (T) of thermovalve type to record the temperature of the process.

EVAPORATION OF ETHANOL-WATER MIXTURE-I

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vi) Vacuumgauge (VG) with range 760 to 0 mm of Hg.vii) Peri-Staltic type variable flow liquid pump (LP) having range up to 100 lt/hr.

viii) Oil diffusion vacuum pump (VP). A vacuum of 650 mm of Hg can be

attained in the apparatus.

Diameter of column 80 mm (3 in)

Packing nominal diameter 8 mm (0.375 in)

Packed height 1.5 m (5 ft)

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PROCEDURE

1. A calibration curve was plotted between LFM readings and liquid flow rate (lt/hr),as

shown in Figure 5.2.

2. Air Flow meter was calibrated using a pre-calibrated flow-meter and confirmed by

the collection of air in a polyethylene bag for a fixed time. A calibration curve was

drawn between air-flow rate and vacuum gauge readings, as shown in Figure 5.3.

3. To measure liquid mixture composition a calibration curve was drawn between liquid

mixture density (gm/cc) and Vol% Ethanol using Anton Parr DMA-48 fluid density

measurement apparatus, as shown in Figure 5.4.

4. 1200 ml of mixture of a fixed composition, say 42% Ethanol by volume was taken in

each experiment with different air flow rates (420 lt/hr to 1220 lt/hr).

5. In each experiment the plant was operated for one hour with liquid flow rate of 6

lt/hr.

6 The experiments were repeated for 30.5, 26.5, 20.5, and 16.5 Vol% Ethanol in

mixtures.

7 In each experiment the column was first dried by passing dry air for 1/2 hr. The feed

liquid was then run through the column for 15 min. Then liquid is allowed to

accumulate in the calibrated liquid leg for 5 min and the liquid level is recorded and

its composition is measured by taking a small sample.

8 The change in liquid level giving the volume evaporated of liquid mixture and

change in composition was recorded after 1hr operation.

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1.00

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0.00 20.00 40.00 60.00 80.00 100.00

10.00 30.00 50.00 70.00 90.00

VOL% ETHANOL

0.80

0.84

0.88

0.92

0.96

0.82

0.86

0.90

0.94

0.98

DENSITY(gm/cc)

Figure 5.4. Density of ethanol-water mixture vs Vol% ethanol in mixture.

T bl 5 1 D t f th ti f i t f i l th l d t

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Table 5.1. Data for the evaporation of mixture of commercial ethanol and waterAir Flow Rate

(lt/hr)

Volume

of Mixture

Evaporated(ml/hr)

Vol% Ethanol

Initial

Vol% Ethanol

after 60 min

Vol% of Mixture

after 60 min

Process Temperature

oC

420 84 42.0 36.0 6.0 27.5

680 105 42.0 35.5 6.5 27.0

940 120 42.0 35.0 7.0 29.0

1220 140 42.0 34.0 8.0 27.5

420 65 30.5 27.0 3.5 27.0

680 85 30.5 25.5 5.0 28.5

940 90 30.5 25.5 5.0 27.0

1220 100 30.5 25.0 5.5 27.0

420 60 26.5 23.5 3.0 28.0

680 70 26.5 23.0 3.5 27.0

940 90 26.5 22.5 4.0 29.0

1220 90 26.5 21.5 5.0 27.5

420 55 20.5 18.5 2.0 29.0

680 65 20.5 17.0 3.5 27.5

940 80 20.5 17.0 3.5 27.0

1220 85 20.5 15.0 5.5 28.0

420 50 16.5 14.7 1.8 27.0

680 55 16.5 14.5 2.0 26.0

940 60 16.5 14.5 2.0 24.0

1220 60 16.5 14.0 2.5 24.0

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MATHEMATICAL MODELING ( An Analytical Approach)

To develop different mathematical models a kinetic approach is

applied to the rate of evaporation of volatile liquids [73]. Due to

universal nature of the Arrhenius Law as it is applied to biological and

medical fields we have used it to study the rate of evaporation and

separation of mixture of volatile liquids. Different models have been

made by using different sets of assumptions. In each model a

mathematical equation is made that contains a parameter K (samesymbol K and k are used for different models, they should be

considered as different for every model). The value of this parameter is

then found by testing it against the experimental data. If the parameter

has the same value for each data, then we say that the fit is good, and

the particular model is also good. However, if different experimentalresults give different values of K, then the model is not considered to

be good.

KINETIC MODEL -IA

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This model is made with the following assumptions [74]:

i) Due to the lower volatility of water as compared to ethanol,evaporation of water is neglected .

ii) Rate of evaporation of ethanol is proportional to itsconcentration, i.e. exponential decrease in concentrationof ethanol as a function of time.

This results in the equation : C(t) = C(0) , where K [1/hr] is thevelocity constant.

or (5.1a)

Values of 'K' are calculated for different air flow rates and feedconcentrations and as there is variation in the values of 'K',

therefore an average value is reported for different air flow ratesin Table 5.2 for ethanol water mixture. The values of Kavgincreases from 0.122 to 0.219 for air flow rate 420 to 1220 lt/hrfor Model-IA. Experimental and calculated values of vol%ethanol in product are reported in Table 5.3.

Kt

e

Kt

C

C t

1 0log

( )

( )

Table 5.2. Values of average rate constants Kavgfor different air flow rates and vol%

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g

ethanol in feed for Model-IA.

Air flow rate

(lt/hr)

420 680 940 1220

Kavg 0.122 0.161 0.168 0.219

Table 5.3. Experimental and calculated values of vol% ethanol in product(Model-IA).

Air Flow Rate

(lit/hr)

Vol% Ethanol

in feed C(0)

Vol% Ethanol

after 60 min

C(t)expt

Vol% Ethanol

after 60 min

C(t)cal

% error

420 42.0 36.0 37.176 -3.270

680 42.0 35.5 35.754 -0.716

940 42.0 35.0 35.505 -1.443

1220 42.0 34.0 33.740 +0.765

420 30.5 27.0 26.997 +0.011

680 30.5 25.5 25.964 -1.820

940 30.5 25.5 25.783 -1.110

1220 30.5 25.0 24.501 +1.996

420 26.5 23.5 23.456 +0.187

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Maximum error in this model is 9.787% which is much below the allowable value of 25%

[76]. Graphical represention of experimental and calculated vol% ethanol in product as a

function of vol% ethanol in feed and air flow rate are shown in Figures 5.5 and 5.6.

Table 5.3 continued

680 26.5 23.0 22.559 +1.917

940 26.5 22.5 22.402 +0.436

1220 26.5 21.5 21.288 +0.986

420 20.5 18.5 18.146 +1.914

680 20.5 17.0 17.452 -2.659

940 20.5 17.0 17.330 -1.941

1220 20.5 15.0 16.468 -9.787

420 16.5 14.75 14.620 +0.881

680 16.5 14.5 14.050 +3.103

940 16.5 14.5 13.950 +3.793

1220 16.5 14.0 13.250 +5.357

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16.00 26.00 36.0021.00 31.00 41.00

Vol% Ethanol in Feed

13.00

18.00

23.00

28.00

33.00

38.00

15.50

20.50

25.50

30.50

35.50

Vol%E

thanolinP

roduct

Air flow rate (lt/hr)

1220 lt/hr

1220 lt/hr

940 lt/hr

940 lt/hr

680 lt/hr

680 lt/hr

420 lt/hr

420 lt/hr

Markers represent Exptl. data.

Curves represent calc. values.

Figure 5.5. Experimental and calculated values of Vol% ethanol in product vs Vol%ethanol in feed for Model-IA.

140

400.00 600.00 800.00 1000.00 1200.00500.00 700.00 900.00 1100.00

Air Flow Rate (lt/hr)

13.00

18.00

23.00

28.00

33.00

38.00

15.50

20.50

25.50

30.50

35.50

Vol%EthanolinProduct

Figure 5.6. Experimental and calculated values of Vol% ethanol in productvs Air flow rate for Model-IA.

Markers represent Exptl. data.Curves represent Calc. values.

141

Vol% ethanol in feed

42%

42%

30.5%

26.5%

26.5%

20.5%

20.5%

16.5%

16.5%

KINETIC MODEL IB

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KINETIC MODEL-IB

i) Due to the lower volatility of water as compared to ethanol,

evaporation of water is neglected.

ii) Linear decrease in concentration of ethanol as a function of timeis assumed i.e.,

C(t) = C(0) - K t

or (5.1b)

where K [hr-1] is a velocity constant. Values of 'K' are calculated fordifferent air flow rates and feed concentration and average values of

'K' are calculated for every air flow rate.

The values of Kavg increases from 3.25 to 5.3 when air flow rate

increases from 420 to 1220 lt/hr for Model-IB.

Maximum error in this model is 20% which is below the allowable

value of 25%

KC(0) C(t)

t

KINETIC MODEL II

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KINETIC MODEL-II

The following four assumptions are used :

i) Due to the lower volatility of water as compared to ethanolevaporation of water is negligible i.e., Vw(t) = Vw(0) = Vw =

constant, where Vwis the volume of water in mixture.ii) Volume of ethanol in mixture is a function of time, i.e. Va(t).

iii) Volume fraction of ethanol (5.2)

iv) From Arrhenius Law, rate of decrease in volume of mixture isproportional to ethanol concentration.Therefore: (5.3)

(5.10)

Again values of 'K' are calculated for different air flow rates and feed

concentration, and as there is variations in the values of 'K' an averagevalue is calculated for every air flow rate, as reported in Table 5.6. Thevalue of Kavg increase from 264 to 405 for the Model-II when air flowrate increases from 420 to 1220 lt/hr.

Maximum error in this model is 17.65% which is below the allowablevalue of 25% [76]. A large positive error due to the reason that aconstant volume of water (VW) is assumed.

Ca tVa t

Vw Va t( )

( )

( )

dVa t

dtKC

at K

Va t

Vw Va t

KVa loss

t

V

t1

Va loss

Va

w log

( ) ( ) ( )

( )

0

0

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KINETIC MODEL -III

This model is based on the following two assumptions.

i) Due to the lower volatility of water as compared to ethanol evaporation ofwater is neglected.

ii) dV/dt = -k [C(t)] is assumed, where dV/dt = volumetric rate of

evaporation of mixture, C(t) = C(0) - Kt, where K [1/hr] is rate constant, n is order of

reaction and k = proportionality constant which is a function of air flow rate,

temperature, liquid flow rate, interfacial area and nature of the liquid.Therefore (5.11)

(5.16)

Where postscript I and J represents two different cases of volume evaporated (V),

vol% ethanol in feed (F), vol%ethanol in product (P) and rate constant (K).

The average values of ns varies from 0.41 to 0.87 for air flow rates from 420 to

1220 lt/hr.

dV

dtk C(t) k [C(0) -Kt]n n [ ]

VI

VJ

KJ

KI

FI PI

FJ PJ

n 1 n 1

n 1 n 1

n

MODELS IVA AND IVB

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MODELS IVA AND IVB

These models are based on the following assumptions :

i) Due to the lower volatility of water as compared to ethanol evaporation of wateris neglected.

ii) (5.17)

where, dV/dt is volumetric rate of evaporation, and n is order of reaction.

C(t) = C(0) , K is rate or velocity constant, and k is proportionality constantas defined in model - III.

To use computer for iterative calculations the above equation is written as under:

MODEL-IVA

Where postscript I and J represents two different cases of ethanol concentration in feed(C), rate constant (K) and volume evaporated (V).

As nK 0 e-nK = 1-nK

Eqn. (5.20) gives:

(5.21)

For iterative computer calculations the above equation is written as under:

MODEL-IVB

For different air flow rates anavgvaries from 0.396 to 0.920 andbnavgvaries from 0.404

to 0.829.

dV

dtk C t k [C(0) en

-Kt n [ ( )] ]

Vk C

nK1 1 nK k [C 0

nn

[ ( )][ ] ( )]

0

VI

VJ

CI

CJ

KJ

KI

1 e

1 e

n

n

nKI

nKJ

[ ]

[ ]* *

[ ]

[ ]

VI

VJ

CI

CJ

n

n

Kt

e

DISCUSSION

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DISCUSSION

The maximum error for Model-IA is 9.8%, which is reasonable for such experimentaldata.

The maximum error for Model-IB is 20.0% at low concentration of ethanol i.e., 16.5%.It is clear from Figure 5.8, that this model gives good results from 42 to 20% ethanol infeed.

The maximum error for Model-II is 17.65%. This negative error is due to the additionof a constant quantity Vwin the denominator of Eqn. (5.2). This assumption have moreeffect due to the more evaporation of water at higher air flow rates.

Therefore analytical Model-IA is more appropriate for this process conditions due tolesser error as compared to other two models. Furthermore by looking at Tables 5.8, 5.9and 5.10 it is clear that order of reaction i.e., values of nsincreases with air flow rate inModel-III, IVA and IVB show same behavior. The values of navg for these modelsmatch each other with maximum error of 4%.

The mathematical models developed by analytical techniques do not give a constantvalue of the parameter 'K' and n for different cases due to the following reasons.

i) Error in the measurements of 'volume evaporated' due to large hold up

of column andsmall volume change.ii) There is also temperature variation (i.e., 24 to 29C) which is not taken into

account.

iii) Error due to the Humidity of input air.

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EVAPORATION OF ETHANOL-WATER

MIXTURE-II

INTRODUCTION

There are two general approaches for curve fittings that are distinguishedfrom each other on the basis of the amount of error associated with the data.

First, when the data exhibits a significant degree of error or 'noise' the

strategy is to derive a single curve that represents the general trend of the

data. Approach of this nature is called Least Square Regression.Second, when the data is known to be very precise the basic approach is

to fit a curve or a series of curves that pass directly through each of the

points. This estimation of values between well-known discrete points iscalled Interpolation.

MATHEMATICAL MODELING (An Empirical Approach)

As our data is of first kind, so it is preferred to apply least square regressiontechnique to see the variation of 'volume evaporated' first as a function of

normalized air flow rate; secondly air flow rate as function of feedconcentration; and thirdly changes in ethanol concentration as a function of

air flow rate. The details of these curves which are made by using computersoftware and experimental data of Table 5.1 are given below.

2.80

153

2.80

154

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0.80 1.20 1.60 2.00 2.40 2.80 3.201.00 1.40 1.80 2.20 2.60 3.00

Air Flow Rate (Normalized)

1.00

1.40

1.80

2.20

2.60

1.20

1.60

2.00

2.40

VolumeEva

porated(Normalized)

Vol% ethanol in feed42%

42%

30.5%

30.5%

26.5%

26.5%

20.5%

20.5%

16.5%

16.5%

Figure 6.1. Shows Normalized (Volume of mixture evaporated vs Air flow rate)using straight line fit for different initial concentration of ethanol.

Markers represent data points.Curves represents straight line fit.

0.80 1.20 1.60 2.00 2.40 2.80 3.20

1.00 1.40 1.80 2.20 2.60 3.00

Air Flow Rate (Normalized)

1.00

1.40

1.80

2.20

2.60

1.20

1.60

2.00

2.40

VolumeEvap

orated(Normalized)


42%

30.5%

30.5%

26.5%

26.5%

20.5%

20.5%

16.5%

16.5%

Figure 6.2. Shows Normalized (Volume of mixture evaporated vs Air flow rate) using

Log regression curve fitting for different initial concentration of ethanol.

Markers represent Exptl.data.Curves represent log regression fit.

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Figure 6.6. Shows Volume of mixture evaporated vs Vol% ethanol in feed using Log

regression curve fitting for ethanol-water mixture for different air flow rates.

Markers represent Expt.; data.Curves represent Log regression fits.

16.00 26.00 36.0021.00 31.00 41.00


50.00

70.00

90.00

110.00

130.00

60.00

80.00

100.00

120.00

140.00

Volumeofmixtu

reevaporated(ml/hr)

Air Flow Rates420 lt/hr

420 lt/hr

680 lt/hr

680 lt/hr

940 lt/hr

940 lt/hr

1220 lt/hr1220 lt/hr

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EMPIRICAL EQUATIONS OF TYPE ( V = a + b Log F + e Log C )

From the above figures it is clear that 'volume evaporated' shows logarithmic relationshipwith 'air flow rate' and 'feed concentration of mixture'. On this basis the following empiricalequation is derived.

(6.1)where F is air flow rate, V1volume evaporated, a' and b are constants.

Similarly (6.2)

Where C is the ethanol concentration , V2is volume evaporated, d and e are empiricalconstants.

Combining Eqns. (6.1) and (6.2) we get:

(6.3)

Let a' + d = aThen (6.4)

The values of 'a', 'b', and 'e' are determined by 'Multiple Linear Regression' technique,for which computer program is attached at Appendix-4. The values determined for, a, b, eand V are given as output. These results indicate that more than 90% of the originaluncertainty has been explained by this model i.e. R 0.90 [75].

Forcommercial ethanol empirical equations for volume evaporated of ethanol andethanol-water mixture as a function of 'air flow rate' and 'feed concentration' are givenbelow:

V(ethanol) = -361.431 + 26.7 Log F + 77.3 Log C (6.5)

V(mixture) = -336.7 + 33.95 Log F + 59.23 Log C (6.6)

The three dimensional graphical representation for Eqns. (6.5) and (6.6) are given in Figures6.11 and 6.12 respectively.

V a b Log F1

/

V V V a d b Log F e Log C1 2 ( )/

V a b Log F + e Log C

2

V d e Log C2

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The maximum error is 16.67% which is below the allowable value of 25%

[76]. Graphical representation of experimental and calculated volume of

mixture evaporated (ml/hr) as a function of log of air flow rate and vol%

ethanol in feed arerepresented in Figures 6.13 and 6.14 respectively. Plot

of calculated vs experimental values of volume of mixture evaporated for

this model are shown in Figure 6.15, the fitted line have equation Y =

1.012*X with R = 0.995736 (which means more than 99.58% of

uncertainity has been explained by this model [75]).

2

Fi 6 11

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Fig 6.11

Figure 6.11. Three dimensional graphical representation for Volume of ethanol evaporated as a function of

Air flow rate and Vol% ethanol in feed for empirical Model-I [Eqn. (6.5)].

ETHA

NOLVOLUMEEV

APORATED(ml/hr).

AIR FLOW RATE(lt/hr). VOL% ETHANOL IN FEED

Fi 6 12

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Fig 6.12

MIX

TUREVOLUMEE

VAPORATED(ml/hr)

AIR FLOW RATE (lt/hr)

Figure 6.12. Three dimensional graphical representation for Volume of mixture evaporated as a function of

Air flow rate and Vol% ethanol in feed for empirical Model-I [Eqn. (6.6)].

165 166

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6.00 6.40 6.80 7.206.20 6.60 7.00

Log of air flow rate (lt/hr)

34.00

54.00

74.00

94.00

114.00

44.00

64.00

84.00

104.00

124.00

Volumeofmixtu

reevaporated(ml/hr)


42%

30.5%

30.5%

26.5%

26.5%

20.5%

20.5%

16.5%

16.5%

Figure 6.13. Shows Experimental and calculated values of Volume of mixture evaporatedvs Log of air flow rate for straight line fits for different Vol% ethanol in feed

for empirical Model-I.

Markers represent Exptl. data.Curves represent Calc. values

2.80 3.00 3.20 3.40 3.60 3.802.90 3.10 3.30 3.50 3.70

Log of vol% ethanol in feed

34.00

54.00

74.00

94.00

114.00

44.00

64.00

84.00

104.00

124.00

Volumeofmixtu

reevaporated(ml/hr)

Air flow rates420 lt/hr

420 lt/hr

680 lt/hr

680 lt/hr

940 lt/hr

940 lt/hr

1220 lt/hr

1220 lt/hr

Figure 6.14. Experimental and calculated values of volume of mixture evaporated vsLog of Vol% ethanol in feed for straight line fit for different Air flow

rates for empirical Model-I.


167

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45.00 65.00 85.00 105.00

55.00 75.00 95.00 115.00

Experimental volume of mixture evaporated (ml/hr)

45.00

65.00

85.00

105.00

55.00

75.00

95.00

115.00

Calcu

latedvolumeofm

ixtureevaporated(ml/hr)


Figure 6.15. Calculated values of volume of mixture evaporated vs Experimental valuesof volume of mixture evaporated for empirical Model-I [Eqn. (6.6)].

V(Calc.) = 1.012* V(Exptl.) and R**2= 0.995736for empirical Model-I [Eqn. (6.6)].

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EMPIRICAL EQUATIONS OF TYPE (Y = a XbZc)

We have done some further analysis of data of Table 6.1 using another technique given in

referance [76]. We have used the computer software and following procedure to find the

empirical equations of the type Y = a Xb

Zc

.

(i) Plotted experimental values of volume of mixture evaporated (VE) vs air flow rate (AF).

This will give us four curves one for each ethanol concentration in feed. Fitted the power

exponential equation: VE= a (AF)band found the value of aand bfor each concentration.

An average value of 0.42 is taken for btaken from Figure 6.16.

(ii) Plotted VE / (AF)0.42vs initial concentration (CF). Fitted the power exponential equation:

VE / (AF)0.42 = c (CF)

dagain. The value of c = 0.681, d = 0.643, and

R2(coefficient of determination) = 0.92 was obtained. Which means more than 92% of

uncertainty has been explained by this model [75] taken from Figure 6.17.

(iii) Plotted (VE/ (CF)0.643) vs AF. Fitted the power equation: VE/ (CF)

0.643= e (AF)f. Then

we obtained e = 0.572, f = 0.427, and R2= 0.921, taken from Figure 6.18.

(iv) The equation VC= 0.572 (AF)0.427 (CF)

0.643was obtained.

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In Table 6.2 are shown the initial concentrations of ethanol, and the

volume of mixture evaporated (experimental and calculated by Model-

II) for different air flow rates. As a first approximation since thetemperature are all within five degree Celsius of each other, we assume

that the effect of temperature may be neglected. The volume

evaporated has been related to the air flow rates and initial

concentration, and it has been found that best fit for the data is made

by the equation:

VC = 0.572 (AF)0.427 (CF)

0.643 (6.7)

Where VC= Total Volume Evaporated (ml)

AF = Air flow rate (lit/hr)

CF= Initial ethanol concentration in mixture (Vol %)

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20.00 25.00 30.00 35.00 40.00

22.50 27.50 32.50 37.50


4.00

5.00

6.00

7.00

8.00

4.50

5.50

6.50

7.50

VolumeEvaporated/(AirFlowRate)**0.42

Solid lines represent equations.Markers represents data.

Y=0.681*X**0.643R**2 = 0.916673

Figure 6.17. Plotted Volume evaporated / (Air flow rate)**0.42 vs Vol% ethanol in feed, using Power law fit for the development of empirical Model-II.

169

420.00 620.00 820.00 1020.00 1220.00520.00 720.00 920.00 1120.00

Air Flow Rate (lt/hr)

65.00

85.00

105.00

125.00

145.00

75.00

95.00

115.00

135.00

155.00

Volumeofmixtureevaporated(ml/hr)


42%

30.5%

30.5%

26.5%

26.5%

20.5%

20.5%

Solid lines represent equation.Markers represent exptl. data.

Figure 6.16. Experimental Volume of mixture evapora ted vs Air flow rate for differentVol% ethanol in feed using Power law fit for the development of empirical

Model-II.

168

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420.00 620.00 820.00 1020.00 1220.00520.00 720.00 920.00 1120.00

Air flow rate (lt/hr)

7.00

9.00

11.00

13.00

8.00

10.00

12.00

14.00

VolumeEvaporated/(Vol%

ethanolinfeed)**0.

641

Solid lines represent equation.Markers represernt Exptl. data.

Y=0.572*X**0.427

R**2=0.921

Figure 6.18. Plotted Volume of mixture evaporated / (Vol% ethanol in feed)**0.643 vsAir flow rate using Power law fit for the development of empiricalModel-II.

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The calculated values of the volume evaporated (VC) are given in

Table 6.2 and is plotted against the experimentally observed value of

volume evaporated (VE

) in Figure 6.19. The markers representexperimental values and continuous curve represents calculated values

of V. For ideal data, all points should lie on the 45 line. As seen in

Figure 6.19, most of the data lies about this line, represented by

equation VC= 0.999583 VEwith R2= 0.99738. The percent error in the

calculation of VCis found by:

Percent error = 100*( VE- VC)/ VE (6.8)

which is also given in Table 6.2. It can be seen that the error in Model-

II is always less than about 8.3 %, which shows that the fit is quite

good.

0

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55.00 75.00 95.00 115.00 135.0065.00 85.00 105.00 125.00

Experimental Volume Evaporated (ml/hr)

52.00

72.00

92.00

112.00

132.00

62.00

82.00

102.00

122.00

CalculatedVo

lumeEvaporated(ml/hr)

45 LineMarkers represent Exptl. data.Solid line represents fitted line.

Figure 6.19. Calculated values of volume evaporated vs Experimental values of Volumeevaporated for empirical Model-II.

Y=0.999583*XR**2=0.99738

DISCUSSION

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DISCUSSION

The error observed in calculated values of two models may be due to the following

reasons:

(i) Error in the measurements of "Volume evaporated " is due to large hold up ofcolumn and small volume change.

(ii) There is also temperature variation (i.e., 24 to 290 C) which is not taken into

account.

(iii) Humidity of input air may also be contributing some error.

We feel that the rate of separation is quite small, which can be improved by increasing

the amount of reflux, which can be done by increasing air flow rate, and by having

temperature difference between the liquids and air for condensation. Also, the inlet air

should be bubbled through the product. Air flow rate can be increased manifold by

having co-current flow, but how this will affect the separation factor, remains to beseen. As the maximum error in Model-II is 8.3%, while the error in Model-I is 16.67%.

Therefore empirical Model-II is better than the empirical Model-I.

o

SEPERATION FACTOR OF ETHONOL-WATER

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SEPERATION FACTOR OF ETHONOL-WATER

MIXTURECALCULATION OF SEPERATION FACTOR (semi-empirical) FOR

ETHONOL-WATER SYSTEM

INTRODUCTION

In this chapter we have discussed the calculations of distillation separation factor of

ethanol-water mixture by analytical (thermodynamic), semi-empirical, and

empirical techniques.

Distillation is the combination of two operations evaporation and condensation. For

the calculation of separation factor, only the experimental data of evaporation is

sufficient. Different setups of apparatus Figure 7.3 to Figure 7.18 and process

conditions, such as degree of vacuum (VM), process temperature (PT), room

temperature (RT), column top temperature (CTT), air flow rate (AFR), density of

feed (DF), density of product (DP), volume of mixture evaporated (VW), volume of

product (VP), volume of feed (VF), calculations factors (A1 and A2) and separation

factor (SF) were recorded in Tables 7.3 to 7.18. To decide about the best setup,

separation factor were calculated for different setups and process conditions.

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MATHEMATICAL

The simplest example of batch distillation is a single stage differential

distillation, starting with a still pot, initially full, heated at a constant rate. In thisprocess the vapor formed on boiling the liquid is removed at once from the

system. This vapor is richer in the more volatile component, with the result that

the composition of the liquid product progressively alters. Thus, whilst the

vapor formed over a short period is in equilibrium with the liquid, the total

vapor formed is not in equilibrium with the residual liquid. At the end of the

process the liquid which has not been vaporized is removed as the bottom

product. The analysis of this process was first proposed by Reyleigh [57].

Let M be the number of moles of material in the still and X be the mole

fraction of component A. Suppose an amount dM, containing a mole fraction Y

of A, be vaporized. Then a material balance on component A gives:

Y dM = d(MX) = MdX + XdM

If MP, MF are the number of moles and XP, XF are volume fraction of product

and feed, then after rearrangement and integration of above equation we get:

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(7.41)

The integral on the right-hand side can be solved graphically if the

equilibrium relationship between Y and X is available. In some cases a

direct integration is possible. Thus if over the range concerned the

equilibrium relationship is a straight line of the form Y= mX+C

where m is the slope of the straight line and C is the intercept,

then:

log {MP/MF} = [1/(m-1)] log {[(m-1)XP + C]/ [(m-1)XF + C]}

or MP/MF = [(Y-X)/(YF-XF)] (m-1) (7.42)

logMP

MF

dX

Y XXF

XP

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From this equation the amount of liquid to be distilled in order to

obtain a liquid of given concentration in the still may be calculated and

from this the average composition of the distillate can be found by amass balance.

Alternatively, if the relative volatility may be assumed constant

over the range concerned, then Y = X/[ 1 + (-1)X] can be

substituted in Eqn. (7.42). This leads to the solution:

log{MP/MF}=[1/(-1)]log{XP(1-XF)/XF(1 - XP)}+log {(1- XF)/(1 -XP)} (7.43)

As this process consists of only a single stage, a complete

separation is impossible unless the relative volatility is infinite.

Application is restricted to conditions where a preliminary separation

is to be followed by a more rigorous distillation, where high purities

are not required, or where the mixture is very easily separated.

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STEP-IIIIn this step overall and component volume balance

equations were written.

If VF, VP and VW are the volume of feed, product, and waste

respectively, and

VF = 1200 ml ; then ( 7.48 )

VP = 1200 - VW ( 7.49 )

Now as XF, XP and XW are the volume fraction ethanol in feed,

product and waste respectively and MF, MP and MW are the moles of

ethanol in feed, product and waste, then we can write:

MF = VF * XF * 0.018333 ( 7.50)

and

MP = VP * XP * 0.018333 ( 7.51 )

STEP-IV Writing formula for single stage batch distillation i.e., differential

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STEP IV Writing formula for single stage batch distillation i.e., differentialdistillation [10] we get:

Log (MP/MF) = (1/A) Log {[XP(1 - XF)]/ [XF(1 - XP)]}

+ Log{(1 - XF)/(1 - XP)} ( 7.52 )

rearranging

A = 1+ Log {[XP(1 - XF)]/[XF(1 - XP)]}(Log(MP/MF)

- Log {(1 - XF)/(1 - XP)} (7.53)

Where A is the separation factor:

Separation factor of ethanol-water system has been calculated fordifferent setups of apparatus. The input and output data are attached in Tables7.3 to 7.18. The symbols used in Tables, are not properly defined above are

given as under:VM = Vacuum (mm of Hg)

AFR = Air flow rate (lt/hr)

DD = DP - DF

PT = Process Temperature

RT = Room Temperature

CTT = Column Top TemperatureA1 = Log{ XP(1 - XF) / (XF(1 - XP))}

A2 = Log{ MP(1 - XP) / ( MF(1 - XF))}

A = 1 + ( A1/A2)

SF = A = Separation factor

100.00

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0.80 0.84 0.88 0.92 0.96 1.00

Density of ethanol (gm/cc)

0.00

20.00

40.00

60.00

80.00

Vol

%e

thanol

Figure 7.19. Vol% ethanol in ethanol-water mixture vs density of ethanol.

Markers represents Expt. data.Curve represents fitted polynomial.

V= -2050.4 D**2 + 3148.0 D - 1100

FIG7 3

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FIG7.3

Figure 7.3 Experimental setup for the separation factor study of ethanol-water mixture using packed column,recycling of liquid feed and dried air flow from the liquid surface at room temperature.

FIG7.5

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FIG7.5

Figure 7.5 Experimental setup for the separation factor study of ethanol-water mixture using, slightly heated

(liquid feed and dried air flow) with air flow from the liquid surface.

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TABLE 7.3 INPUT AND OUTPUT DATA SETUP OF

FIGURE 7.3

S.No VM AFR DF DP DD VW PT RT CTT VP XF XP MF MP A1 A2 SF

1 300 420 0.9508 0.9566 0.0058 84 27.5 30 28 1116 0.3951 0.3509 8.6930 7.1792 -0.1894 -0.1207 2.5685

2 200 680 0.9508 0.9576 0.0068 105 27 30 28 1095 0.3951 0.3431 8.6930 6.8881 -0.2237 -0.1502 2.4889

3 100 940 0.9508 0.9581 0.0073 120 29 30 28 1080 0.3951 0.3392 8.6930 6.7166 -0.2410 -0.1695 2.4217

4 0 1220 0.9508 0.9596 0.0088 140 27.5 30 28 1060 0.3951 0.3275 8.6930 6.3636 -0.2939 -0.2059 2.4278

5 300 420 0.9646 0.9680 0.0034 65 27 30 28 1135 0.2876 0.2599 6.3270 5.4080 -0.1394 -0.1188 2.1733

6 200 680 0.9646 0.9695 0.0049 85 28.5 30 28 1115 0.2876 0.2475 6.3270 5.0598 -0.2047 -0.1688 2.2131

7 100 940 0.9646 0.9696 0.0050 90 27 30 28 1110 0.2876 0.2467 6.3270 5.0203 -0.2092 -0.1755 2.1918

8 0 1220 0.9646 0.9703 0.0057 100 27 30 28 1100 0.2876 0.2409 6.3270 4.8580 -0.2407 -0.2007 2.1992

9 300 420 0.9689 0.9721 0.0032 60 28 30 28 1140 0.2525 0.2259 5.5547 4.7206 -0.1464 -0.1277 2.1463

10 200 680 0.9689 0.9725 0.0036 70 27 30 28 1130 0.2525 0.2225 5.5547 4.6097 -0.1657 -0.1472 2.1259

11 100 940 0.9689 0.9732 0.0043 90 29 30 28 1110 0.2525 0.2166 5.5547 4.4082 -0.2001 -0.1843 2.0856

12 0 1220 0.9689 0.9742 0.0053 90 27.5 30 28 1110 0.2525 0.2082 5.5547 4.2363 -0.2506 -0.2134 2.1745

13 300 420 0.9757 0.9779 0.0022 55 29 30 28 1145 0.1954 0.1766 4.2992 3.7061 -0.1247 -0.1253 1.9956

14 200 680 0.9757 0.9789 0.0032 65 27.5 30 28 1135 0.1954 0.1679 4.2992 3.4939 -0.1853 -0.1738 2.0665

15 100 940 0.9757 0.9790 0.0033 80 27 30 28 1120 0.1954 0.1670 4.2992 3.4299 -0.1916 -0.1912 2.0017

16 0 1220 0.9757 0.9817 0.0060 85 28 30 28 1115 0.1954 0.1435 4.2992 2.9332 -0.3714 -0.3198 2.1614

17 300 420 0.9800 0.9819 0.0019 50 27 30 28 1150 0.1584 0.1417 3.4838 2.9883 -0.1304 -0.1339 1.9742

18 200 680 0.9800 0.9824 0.0024 55 24 30 28 1145 0.1584 0.1373 3.4838 2.8830 -0.1671 -0.1646 2.0147

19 100 940 0.9800 0.9824 0.0024 60 26 30 28 1140 0.1584 0.1373 3.4838 2.8704 -0.1671 -0.1690 1.9884

20 0 1220 0.9800 0.9827 0.0027 60 24 30 28 1140 0.1584 0.1347 3.4838 2.8151 -0.1896 -0.1854 2.0224

TABLE 7 5 Input and Output Data For Setup of Figure 7 5

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TABLE 7.5 Input and Output Data For Setup of Figure 7.5

S.No VM AFR DF DP DD VW PT RT CTT VP XF XP MF MP A1 A2 SF

1 400 145 0.9529 0.9552 0.0023 20 30 12 23 1180 0.3793 0.3617 8.3441 7.8247 -0.0754 -0.0363 3.0746

2 350 270 0.9529 0.9564 0.0035 60 30 12 23 1140 0.3793 0.3524 8.3441 7.3659 -0.1157 -0.0824 2.4052

3 300 420 0.9529 0.9576 0.0047 75 30 12 23 1125 0.3793 0.3431 8.3441 7.0768 -0.1568 -0.1081 2.4504

4 200 680 0.9529 0.9593 0.0064 95 30 12 23 1105 0.3793 0.3298 8.3441 6.6816 -0.2164 -0.1455 2.4868

5 100 940 0.9529 0.9614 0.0085 100 30 12 23 1100 0.3793 0.3132 8.3441 6.3167 -0.2925 -0.1772 2.6502

6 350 270 0.9605 0.9657 0.0052 55 30 12 23 1145 0.3204 0.2787 7.0479 5.8500 -0.1989 -0.1268 2.5689

7 300 420 0.9605 0.9662 0.0057 80 35 12 23 1120 0.3204 0.2746 7.0479 5.6388 -0.2192 -0.1579 2.3880

8 250 540 0.9605 0.9654 0.0049 70 25 12 23 1130 0.3204 0.2811 7.0479 5.8238 -0.1868 -0.1346 2.3874

9 200 680 0.9605 0.9664 0.0059 90 30 12 23 1110 0.3204 0.2730 7.0479 5.5553 -0.2274 -0.1706 2.3329

10 150 810 0.9605 0.9678 0.0073 100 30 12 23 1100 0.3204 0.2615 7.0479 5.2743 -0.2859 -0.2069 2.381911 350 270 0.9687 0.9723 0.0036 50 35 12 23 1150 0.2541 0.2242 5.5910 4.7267 -0.1647 -0.1286 2.2813

12 300 420 0.9687 0.9743 0.0056 90 40 12 23 1110 0.2541 0.2073 5.5910 4.2190 -0.2645 -0.2207 2.1984

13 250 540 0.9687 0.9745 0.0058 80 35 12 23 1120 0.2541 0.2056 5.5910 4.2222 -0.2748 -0.2178 2.2618

14 200 680 0.9687 0.9746 0.0059 85 30 12 23 1115 0.2541 0.2048 5.5910 4.1860 -0.2800 -0.2253 2.2427

15 150 810 0.9687 0.9743 0.0056 85 25 12 23 1115 0.2541 0.2073 5.5910 4.2380 -0.2645 -0.2162 2.2233

16 350 270 0.9746 0.9783 0.0037 60 30 12 23 1140 0.2048 0.1731 4.5051 3.6178 -0.2071 -0.1803 2.1489

17 300 420 0.9746 0.9792 0.0046 70 35 12 23 1130 0.2048 0.1653 4.5051 3.4246 -0.2626 -0.2258 2.1629

18 250 540 0.9746 0.9798 0.0052 70 30 12 23 1130 0.2048 0.1601 4.5051 3.3167 -0.3008 -0.2516 2.1957

19 200 680 0.9746 0.9790 0.0044 85 30 12 23 1115 0.2048 0.1670 4.5051 3.4146 -0.2500 -0.2308 2.0835

20 150 810 0.9746 0.9797 0.0051 95 35 12 23 1105 0.2048 0.1610 4.5051 3.2609 -0.2944 -0.2696 2.0920

21 350 270 0.9795 0.9832 0.0037 60 40 12 23 1140 0.1627 0.1303 3.5795 2.7229 -0.2602 -0.2355 2.1048

22 300 420 0.9795 0.9836 0.0041 65 30 12 23 1135 0.1627 0.1267 3.5795 2.6373 -0.2918 -0.2634 2.1079

23 250 540 0.9795 0.9840 0.0045 70 35 12 23 1130 0.1627 0.1232 3.5795 2.5522 -0.3243 -0.2921 2.1099

24 200 680 0.9795 0.9845 0.0050 75 30 12 23 1125 0.1627 0.1188 3.5795 2.4493 -0.3660 -0.3282 2.1151

25 150 810 0.9795 0.9850 0.0055 80 25 12 23 1120 0.1627 0.1143 3.5795 2.3471 -0.4093 -0.3659 2.1186

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DEVELOPMENT OF EMPIRICAL RELATIONSHIP FOR

SEPERATION FACTOR

DEVELOPMENT OF COMPUTER PROGRAM

Computers programs have been written in FORTRAN to use the data obtained in the

experimental results reported in Tables 7.3 to 7.18. Separation factors (empirical) are

calculated by using the models of Eqns. (7.54), (7.55), (7.56) and (7.57) by using

computer programs mod-5 and mod-6 (modified form of mod-4) developed insection 6.2.1.

Mod-5 calculates the empirical constants of Eqns. (7.54), (7.55) and (7.56) with

two independent variables, where as mod-6 calculates the empirical constants of Eqn.

(7.57) with three independent variables.

Different combinations of last three columns of master input file are used taking

two at a time along with first two columns. For example column 3 and 4 are used forEqn. (7.54), column 3 and 5 for Eqn (7.55) and column 3 and 4 for ARF=0 for Eqn.

(7.56) for vacuum distillation. For Eqn. (7.57) all the three columns of input file are

used. The steps involved in this are given in the next slide.

STEP I Input data from Tables 7 3 to 7 18 are merged to a single file and an

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STEP-I Input data from Tables 7.3 to 7.18 are merged to a single file and an

master input file is prepared for 371 points, each data point having been taken

after 1.5 hr of operation time.

STEP-II A computer code called MOD-5 has been developed to calculate thevalues of constants of the following empirical equations:

SF = a + b Log (DF) + e Log (T) (7.54)

SF = a + b Log (DF) + e Log (AFR) (7.55)

SF(VD) = a + b Log (DF) + e Log (T) (7.56)

Where AFR is the air flow rate in lt/hr, and T is the temperature in K. Eqn.(7.56) is for vacuum distillation , in this case AFR = 0.

STEP-III A computer code called MOD-6 has been developed to

calculate the values of constants for empirical equation of the type:

SF = a + b Log (DF) + e Log (AFR) + d Log (T) (7.57)

The computer programs, Input and Output files for these models are

attached at Appendix-5 and 6.

RESULTS

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Resulting empirical equations for different cases are given below:

CASE-I SF = f (DF, T)

SF = 0.539 - 10.403 Log (DF) + 0.236 Log (T) (7.58)

DF(0.9492 gm/cc to 0.9969 gm/cc), T(294.5 K to 350 K)

CASE-II SF = f (DF, AFR)

SF = 1.797 - 10.466 Log(DF) +0.014 Log (AFR) (7.59)

DF(0.9492 gm/cc to 0.9969 gm/cc), AFR(25 lt/hr to 1220 lt/hr)

CASE-III SF (VD) = f (DF, T)

SF = -12.746 - 20.494 Log (DF) + 2.506 Log (T) (7.60)

DF(0.9810 gm/cc to 0.9944 gm/cc), T(314 K to 350 K)

CASE-IV SF = f (DF, AFR, T)

SF = 3.008 - 10.258 Log (DF) + 0.016 Log (AFR) - 0.214 Log(T) (7.61)

DF(0.9492 gm/cc to 0.9969 gm/cc), AFR(25 lt/hr to 1220 lt/hr), T(294 K to 335 K)

(i) Values of SF (Experimental) given in Tables 7.3 to 7.18 and SF (Calculated) by this program

are plotted in Figures 7.20 to 7.23. Fitted straight lines through origin have slopes almost equal to

1 i.e., 450 showing that the calculated values are in good agreement with the experimental values.

(ii) Surface plots of Eqns. (7.58), (7.59) and (7.60) are also shown in Figures 7.24, 7.25 and 7.26.

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1.80 2.00 2.20 2.40 2.601.90 2.10 2.30 2.50

Separation Factor (experimental)

1.80

2.00

2.20

2.40

2.60

Separation

Factor(calculated)

SF = f (DF, PT)

Y = 0.9995*X

R**2 = 0.9995

Figure 7.20. Calculated vs Experimental separation factor as a function of density of feedand process temperature.

Markers represents Calculated (SF) vs Experimental (SF).Straight line represents fitted straight line through origin.

1.80 2.00 2.20 2.40 2.601.90 2.10 2.30 2.50


1.80

2.00

2.20

2.40

2.60

1.90

2.10

2.30

2.50

SeparationFactor(calculated)

R**2 = 0.9995

SF = f (DF, AFR)

Y = 0.999999*X

Figure 7.21. Calculated vs Experimental separation factor as a function of density of feedand air flow rate.

Markers represents Calculated (SF] vs Experimental (SF).Line represents fitted straight line through origin.

197

199

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Markers represents Calculated (SF) vs Experimental (SF).Line represents fitted straight line through origin.

SF = f (DF, AFR, PT)

Number of data points = 293

Y = 0.999552*X

R**2 = 0.999539

Figure 7.23. Calculated vs Experimental separation factor as a function of density of feed, air flow rate and process temperature.

199

1.80 2.00 2.20 2.40


1.80

2.00

2.20

2.40

SeparationFa

ctor(calculated)

SF(VD) = f (DF, PT)

Y = 0.999961*X

R**2 = 0.999812

Markers represents Calculated vs Experimental Separation factor.Straight line represents straight line fit through origin.

Figure 7.22. Calculated vs Experimental separation factor as a function of density offeed and process temperature for vacuum distillation i.e. no air flow.

1.95 2.00 2.05 2.10 2.15 2.20 2.25


1.90

2.00

2.10

2.20

2.30

SeparationF

actor(calculated)

Fig 7.24

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Fig 7.24

0.95

0.96

0.97

0.98

0.99300

320

340

2

2.2

2.4

0.95

0.96

0.97

0.98

0.99

Figure 7.24. Surface plot of separation factor as a function of ethanol-water mixture feed density and

process temperature for empirical model MOD-5.

Density of Mixture (gm/cc)

Process Temperature ( K)Separation

Facto

r(calculated)

0

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FIG 7.25

0.95

0.96

0.97

0.98

0.99

250

500

750

10002

2.2

2.4

0.95

0.96

0.97

0.98

0.99

Figure 7.25. Surface plot of separation factor as a function of ethanol-water

mixture feed density and air flow rate for empirical model MOD-5.

Density of Mixture (gm/cc)Air Flow Rate (lt/hr)

Separation

Factor

(calculated)

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CONCLUSION AND SUGGESTION

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CONCLUSIONS

Separation of ideal (isotopic) liquid mixture e.g. H2O-HDO-D2O is

possible only due to mass difference (which gives rise to zero point

energy difference) at low temperature, because the Van der Waals fields

are almost equal in these systems. As the zero point energy (vibrational

energy) of lighter molecule is more as compared to the heavier molecule,

therefore lighter molecule needs less energy for detaching it from other

liquid molecules and this proves that normal effect prevails at low

temperature. Now as the temperature is raised the infrared frequency of

lighter molecule is absorbed more strongly by heavier molecule and as a

result it needs lesser energy for evaporation and becomes more volatile,

therefore inverseeffect prevails at higher temperature e.g.,

in case of H2O-HDO-D2O system this inversion temperature is 230C.

Th th ti l t d hi h i l d th i f lit t f 1933

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The theoretical study which includes the review of literature from 1933

to 1998 regarding isotopic exchange reaction H2O + D2O 2HDO

reveals that theoretical calculations of equilibrium constant for thisreaction are easier in vapor phase than in liquid phase due to the

involvement of Van der Waals forces in liquid phase in general cases,

but cancel out in case of isotopes. We also note that:

(i) Kl Kv at lower temperatures and Kl Kv at higher

temperatures.

(ii) In the theoretical derivation of relationship between K and T it is

seen that if harmonic molecular spectroscopic data is used the value ofK is more near to the classical value of 4.0, and if anharmonic data is

used the values are much smaller than 4.0.

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As the theoretical study of separation factor of ethanol water mixture

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As the theoretical study of separation factor of ethanol-water mixture

is very difficult due to non-ideality of this system therefore an

experimental study was conducted. Firstly the evaporation of ethanol-

water mixture was studied. In the first step four mathematical

(analytical) models were developed to calculate volume evaporated

and their validity was checked by fitting experimental data. In case of

analytical Model-IA the maximum error in the calculated volume %

ethanol in product is approximately 10%, in analytical Model-IB the

maximum error in the calculated volume % ethanol in product is 20%

and in case of analytical Model-II the maximum error in calculated

volume evaporated is 17.65%. Although these errors are less than the

acceptable value of 25% [74], still we have tested two more empirical

models.

Two empirical models were also developed and tested for this data in

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Two empirical models were also developed and tested for this data, in

case of empirical Model-I the maximum error in calculated volume

evaporated is 16.67%, while in case of empirical Model-II themaximum error is only 8.3%, which is lowest of all the tested models.

For the experimental separation factor studies of ethanol-water system,

sixteen different setups of apparatus and different process conditions

are used to collect the separation factor data. Approximately 370 data

points are collected. These data points values are firstly used in

Relayigh`s analytical equation for the calculation of separation factor

for each data point [55]. The separation factor values for ethanol-water

mixture for these process conditions are very much near to the

experimental value of 2.5.

Four different empirical relationships are also developed by using the above

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p p p y g

separation factor (experimental) values and two and three different independent

variables. The first model SF = f (DF, T) has goodness of fit value = 0.864, the

second model SF = f (DF, AFR) has goodness of fit value = 0.875, the third

model for vacuum distillation SF (VD) = f (DF, T) has goodness of fit value =

0.883, the fourth model SF = f (DF, AFR, T) has goodness of fit value = 0.880.

Although the value of co-efficient of determination is slightly less in case of

fourth model as compared to third model, but the number of data points are muchmore in case of 4th model, so the best model which has three independent

variables is the fourth model and this model is valid for the following ranges of

three different variables, DF (0.9492 gm/cc to 0.9969gm/cc), AFR (25 lt/hr to

1220 lt/hr), T (294 to 335 K). When calculated vs experimental separation factor

values of this model are plotted we get a straight line with slope nearly equal to 1

with goodness of fit value = 0.99954.

0

SUGGESTIONS FOR FUTURE WORK

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Reasons for the following experimental facts should be investigated.

(i) Why at low temperatures i.e. up to 25C, Kl Kv and at higher

temperatures i.e. 25C, Kl Kv.

(ii) Why K (HM) K (ANH).

(iii) As we also know that Kl Kv, does it indicate

any relevance between (i) and (ii).

(iv) Why heavy water distillation plants are not operated at higher

temperature to have more D2O in the vapor phase as at higher temperaturesinverseeffect prevails. A study should be conducted for the separation of

heavy water at higher pressure and temperatures (more than 230C).

(v) As and ( ) is maximum at 170C, a study should be

conducted at this temperature for the separation of heavy water from light

water.

P

P

H O

D O

2

2

P PH O D O2 2

Experimental results regarding ethanol-water separation can be improved by

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taking the following measures.

(i) Measurement of volume evaporated can be made more exact byreducing the quantity of hold-up volume.

(ii) Process temperature variations should be made minimum.(iii) Humidity control of evaporating air should be improved.

The rate of separation can be improved by increasing the amount of reflux,

which can be achieved by increasing temperature difference between theprocess temperature and cooling water temperature also the inlet air should be

bubbled through the liquid. Air flow rate can be increased manifold by having

concurrent flow, but how this will affect the separation factor remains to beseen.

A power law fit be done for four models by relating Log (SF) to Log (AFR),Log (DF) and Log (T). Although the value of goodness of fit is maximum for

VD, but the air flow is zero in this case. The introduction of air is giving somenegative effect on the separation of mixture due to the formation of a thin layer

of air between vapor and liquid films. So this process can be used for theliquid mixtures separation at low temperature if we have some way of

separating air-water mixture after helping the first evaporation of the liquid

mixture.

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