EDDY.DIFFUSION COEFFICIENT IN VALVE TRAY DISTILLATION COL UIVINS

EDDY.DIFFUSION COEFFICIENT IN VALVE TRAY DISTILLATION COL UIVINS

By

K. J\IOL'L.\.R

Department of Chemical :\lachilleries and Agricultural Industrie~

Technical University. Budap~est (Received April 2. 1974)

Presented by Prof. Dr. S. SZE"'TGYORGYI

Introduction

For separation processes, mostly equipment of column type is applied providing intensive mass transfer. In tray distillation columns, material trans- fer occurs between the steam and liquid phases. As pointed out earlier [1], hesides the classical design of bubble cap trays, combined tray structures like yah-e trays are now increasingly adopted for distillation columns. These tray structures show, howeyer, a significant degree of weeping affecting in turn, the conditions of mixing on the tray, and also the eddy-diffusion coefficient, characteristic of these conditions. The eddy-diffusion coefficient has to ^}w known to determine the number of trays actually required for the separation of a gIven sharpness.

~fixing of liquid on a tray

A certain degree of liquid mixing occurs on the tray of a distillation column. Assuming a complete mixing, liquid concentration is uniform all oyrr the tray, i.e. the so-called lVIurphree tray efficiency is identical with point efficiency [2]. In the other limiting case no liquid mixing occurs at all, this heing the so-called plug flo'w 'where there is a clean-out correlation between tray efficiency and point efficiency [3]. In practice, some liquid mixing occurs hetween these limiting cases. The most universally accepted model for descrih- ing the mixing of liquid is the so-called eddy-diffusion model [4,5], according to which the correlation hetween tray efficiency and point efficiency IS

1 - e-(7J+ Pe ) e" - 1

171

^{1 .J.... I) 1}

I) Pe

(1)

(I) Pe)

(1

.J.... - - ' - - - -

where

156

" mG\J

I.

= - - ' -

Pe

L,V1

vZ₁ D

K. JfOL.V~[R

For determining the tray efficiency, the eddy-diffusion codficient must he known.

Since the eddy-diffusion coefficient depends only on the flow character- istics, it is influenced to a certain extent hy the tray design, decisive for the flow conditions.

Several correlations for calculating the eddy-diffusion coefficient are known from the literature, hut these apply only to certain experimental tray designs.

For huhhle cap trays, the following correlation has heen suggested hy GERSTER et al. [4, 5]:

DO,5 = 0.00378

+

O.OI71uG

o

^.00I02L*

+

0.000I758h,v . (2) Eq. (2) is widely used within the specific range of application. Many general works on this suhject also suggest its application [6, 7].

Examining the conditions ofliquid mixing on a sieve tray column, Barker and Self have suggested the folIo'wing equation [8]:

where

The correlation suggested for sieve trays hy Foss, GERsrER and PIGFORD [9] is:

----'=--

_ [Zc V

Cl - -

^]-C,

V3Z_c Zl ^J

"where Cl and C₂ are constants, V

=

rate of foam flow.

Owing to the differences in tray design, these correlations do not apply to valve trays. Weeping through the chimney openings of valve trays affects hoth the mixing conditions on the trays and the levels of clear liquid and foam.

This study deals "with an experiment at 'work for estahlishing a correlation suitahle for calculating the eddy-diffusion coefficient for valve trays.

The phenomenon of weeping rcquires the elahoration of a new model different from that used hy Gerster, Barker and Self. In the ne"w model, the discretely arranged catch-holcs are taken into consideration hetween the inlet on the tray and the outlet gate.

An approximative model has heen produced [1, 10], facilitating analyt- ical and grapho-analytical determination of the eddy-diffusion coefficient.

EDDY-DIFPC:HO.v COEFFICIKYT 157

Analyzis of the steady-state concentration profile has heen adopted for the tests. The parameters of liquid mixing were determined by analyzing the concentration profile of a so-called indicator "which did not take part in the lllass transfer process. 'Writing the differential material halance of the indicator as a dissol...-ed matter for the differential section of the tray [1, 10],

d^z-o

_,_1_ (LMbe _ Slvl

z)

^{dct: . 0}

AQ!vl Z/ dz

(3)

introducmg -:.he dimensionless locus co-ordinatpw and performing certain modifications :

Soh-ing differential equation (4) and rearranging:

where

b* - wa*

If!

= ..

b* a*

i) (1 w)

Ib*

a* = a D=lCS b* - b D lC L

*

a (1

2 w)]

le dimensionless locus co-ordinate.

(4.)

(5)

With the knowledge of the concentration profile of the indicator, eddy-diffu- sion coefficient D is ohtained hy Eq. (5) for the gi...-en operating parameters.

The steady-state concentration profile as test method

This test method essentially consists of the following steps: the indicator solution is injected into the liquid flow through an injecting network near the outlet gate; maintaining the rate of injection at a constant level, distrihution of the indicator concentration is estahlished against the direction of flow.

158 K . .1lOLYAU

Under steady operating conditions, the concentration profile does not vary.

It can be assumed here that a one-dimension diffusion process takes place on the tray of a tray column between the inlet and outlet gate.

Knowing the concentration profile of the indicator, the eddy-diffusion coefficient corresponding to tllf' extent of mlxmg on the tray can be estab- lished.

Description of test equipment

Our tests were performed in a dia. 400 mm plexiglass column, illustrated in Fig. 1. Two valve trays and a double-bottom tray of special design were incorporated in the column.

Function of the double-bottom tray was to discharge tll{' liquid ,,·eeping from the measuring tray above via an external hydraulic seal.

The tests were carried out in a water-air system. Supply tank marked 4 was constantly supplied with frE'sh water from the supply mains. \Vatt'r was fed directly to the inlct segment of the tray locatt'd above the measuring tray.

ThE' level of the dear liquid established on the tray was measured at several points between inlet and outlet by means of level gauges (Zd.

Air was delivered into the test equipment by the fan marked 7. Rates of flow of air and water could bc controlled as shown in Fig. 1. The quantity of liquid weeping through the tray was determined by volume.

The measuring tray is illustrated in Fig. 2. Height of the outlet gate could be varied between hw 25 to 80 mm. A constant static hydraulic seal of 15 mm height was maintained during every test. In order to provide iden- tical number of caps in every row, there were also half vaIn's on the tray.

The uSe of half caps is a common method [4]. The injecting Ilf'twork wail mounted on the measuring tray 10 mm away from the outlet gate, as shown in Fig. 2. Position of the injecting network could he varied to suit thE' hE'ight of outlet gate.

j\" umber of sampling rows were pointed out on tll!' measuring tray, and the concentrations for the individual rows were obtained by calculating the average of samples taken at several points of each row.

The valve caps used for our tests were of Glitsch type illustrated in Fig. 3.

The indicator solution was prepared in a tank marked 3 in Fig. 1, and transferred into charging tank marked 2. The indicator solution was delivered to the injecting network by a screw pump; the flow rate was controllable continuously at a constant level.

The indicator solution contained sodium chloride, since brine does not take part in mass transfer bet"ween the gas and liquid phases, and its con- centration can be determined from its conductivity. Concentration and quan- tity of the brine injected in our tests were established to rcsult in a concentra-

EDDY.DIFFUSIOS COEFFICIENT 159

c

160 ^{K . . UOLy..jR}

--

Sample taken ...-"<:D'---'l;;j!:;.!1---...!ti:J'-, at w=l/Row 11

Brine feed

Fig. 2. Design of measutillg tray with injecting network

Row 5 Row 6 Row 7 Row 8

l

^NaCl

tioll not exceeding 2

ou.dm.solution on the tray, hecause the conductiyity of the hrine shows a linear change with concentration up to that limit [10].

The solution samples taken in our tests were thermostated at their original temperature.

1:,·DDY·DIFFL"SIO.Y COEFFICIEST 161

In the tests we measured the effect of gas load, liquid load and gate height on the liquid mixing (eddy-diffusion coefficient) and weeping. Having adjusted a fixed gate height and a fixed liquid load, measurements were takep hy increasing the gas load.

The foHo'wing test ranges were used:

hw

=

25 mm; 40 mm; 55 mm; 75 mm (gate heights) L = 2; 3; 4·; 5 cu.Il1.;hour (liquid loads)

llG 0.546 - 1.29 ^{i l l} sec (linear gas yelocity).

A detailed description of tbp- test equipment and rcsults haye heen puhlish- ed [10].

Test results

A significant degree of weeping was ohseryed on the test traY within the ahove ranges.

The rate of 'weeping is plotted in Figs 4 and 5. Apparently, at a eonstant gate height and liquid load, the quantity of liquid weeping through the tray decreases with increasing gas load. Oln-iously, at loads of e.g. F=

I[~li

^{kg ]}

see eU.In.

and L*

=

8.3 [cu.Ill-mh] the percentage of liquid ,,-eeping through the tray amounts to 31.7 prr eent of the total liquid entering the tray. Although Norman's results [ll] haye been plotted in Fig. 4, the yah-e cap design used by him differed so much from that adopted by us that the comparison applies only to the charaeter of changeo:.

As apparent from Fig. 5, the quantity of the liquid weeping through the tray increased with liquid load.

Based on the literature [I, 10], the eddy-diffusion eorffieient was estah- lished on a model allowing for weeping.

The quantity In If!'. ~ is plotted as a funetion of i) on the 1)a5i8

(

;'C x J

Xg ;to

of Eq. (5) (see Fig. 6). It is apparrnt from the diagram that a straight line can he fitted to hrtween the Illrasurement results, thus verifying the the01'r- tieal model. Based on Eq. (5), the eddy-diffusion coefficient can he deter- mined from the slope of the straight line. A total of 60 test runs were completed in the measuring range specified before; the detailed results haye heen puhlish-

cd [10].

FOl' calculating the eddy-diffusion coefficient, a correlation similar to Eq. (2), and applying to huhhle trays, has heen set up, with the constants estahlished on the principle of parallels and hy the method of least squares.

The final correlation obtained is

DO,5 = 0.0005 0.01285 ^llG

+

0.001755 L* ...L 0.000312 hw (6)

162

Bottom view

K . . \[OLY.-iR

Weight of valve cap' 30 p Lift, max. 5 mm

Fig. 3. Type of experimental valve cap

S*.10-³ ...--.,---.,---r---r---r----;;---;--r--,---;---;--...,

15

10

5

0

0,0 0,5

Fig. 4.

1,0 1,5

Other than Glitsch - type

hw=75mm

co. =2,7 mO/m h + C' =5,5 m3/m h 0

L* =8,5 m³/m h 6.

[40J •

2,0 2,5

F valve cap Rate of weeping vs. gas load factor

3,0

EDDY·DI FFCSIOS COEFFlCIEST

15

10

5

I

o

^~~~~~

__

^~

__

^{L-~ _ _ ~~ _ _ ~~ _ _ ~~}

o

^{2 4 6 8 10 12 14 16 18 20 22 24}

L"

Fig. 5. Rate of weeping vs. liquid load

100 x -Xo 9 8 '¥. x9- x0 7 6 5 4

2

10 9 8

7 6 5 4

,

,

...3>.1

r-^~;,

"

r-

I

I

o

I

1 1

1 1 1

I"

^I 1 ^I

I

~I

t .... ^I

I

I _!

I"''k

1

I

I

1 I

I

I

I

^I

I

I ^{I I}

I

1 I

1 ^{1 1}

I

I I

1

I

I

1

I

I

^I

I I

I I 1

I

'.

I I I

1 1

1

1 1 1 1

I I I

I

I

1

I I

I

I I

I !

I

I I I

I

^I

I

i'\.

N

_I I " I

^{I I}

1

.,..

1 1 _I " I i\.

I

I I ^{I I} I ' I !

I

I

¹

i ^I I 0=0.203.10" m'/sec

I ^I

i

I

! I _I ¹ 1

I

I

^I

i ! i :

4 5 6

'1.10' Fig. 6. Determination of eddy-diffusion coefficient

163

164

where

[llG] = ll1.sec [lzlV] = mm [L*]

=

cu.m.!mh [D]

=

sq.m.sec

K. MOL:lAR

linear gas ydo,:it y gate height

circumferential liquid load eddy-diffusion coefficient.

In Fig. 7 the values obtained for the cddy-diffusion coefficient are plotted as a function of the calculated values. The lines marking deviations of 10 per cent plus and minus are clearly visible on the diagram. The suggested eorre- lation (6) can be stated to describe the points of measurement with a fair aecuracy.

In Fig. 8 our test results are plotted versui' the results ealculated hy

GERSTER et al. [4,5] for bubble trays aeeording to Eq. (:2).

As apparent from Fig. 8, for the range tested and ,,-hen using valve caps, liquid load and gate height influence the eddy-diffusion coefficient to a higher degree than does gas load as oppo:3cd to bubble cap columns.

It is also apparent from Fig. 8 that Eq. (:2) is not valid for valve trays.

For the most common operating parameters it can }H' stated on the strength

D_test 0,5 _{I i} I I ⁱ I i

! i I I , I I II'

006 , r-~I---+,--+I--ri I I I ! ~--+-~'-~~-r~~~-~~~~~~~-~,~~~, iLl

I----l- ^j ..A..'i.-j"1 XI I

f--+

^+, hw =25mm j j J I", 11':' !

r---+-

^{j I i i" i I I}

I ! I I : it I I

0,05!-- .

°

^{hw ='40I11m I i I I I} i),":: 11:: , I 1 i I

t---- i

! - - t:. hw = 55'mm . i , i , 1.- I I ' I

:==

^{IJ hw = 75'mm ! -i i}

^{i !}

^{1 I! I}

4 1

¹

lL1° ,\

^{I I 1} i i

0,0 1--+1---'--'1--:-1 ---r.~f--+-! -+I/---;;/"-+:-; ".11 i! ! i l l i

i i A ~+ !V! I I I i

I YVILi I I1

ⁱ

II!I I

0,03 f---+--+I-~-+~-+r-lL-i!'-.L-Vf-7+I-+-+-+-+l-iI-+-T-1 h-+---

I

HI

--t--i-~!

I

VlLL!

^{I1I i I}

v/v ^{1 1 11 1}

0,Q3 0,04 0,05 0,06

D°,5 = 0,0005 + 0,01285u_G + 0,001755 L" + 0,000312 hw

Fig. 7. Test vs. calculated eddy-diffusion coefficient

0,Q7

EDDY-DIFFUSIOS COEFFICIEST

D,5 0,07

)test ^{I i}

I I

I I 1

1 1 A

0.06

f---t-

-

^{- + hw=25mm I J I}

e -- ri 0

' - - -

-

o hw=40mm , I

I I^U i 0,05

-

^-

0,04

-

-'"

^{hw=55mm 00 I 10 I} _It.It ^{I +}

I 0 hw = 75mm 1 4- I", ⁺ 1/

---r-I

+ " ^,

I .2

I

'"I fd

^l;

I _"'I

'" _{'I /i"}

'2>

I ^d/

^I 1

I

I I +V r ^I

^I

/:

^{I I}

V

¹

0,03

/+ ^I

I I

+7i

^I i

/

^{1 I .1 I}

/

^{I 1}

^{I I}

^{I I}

^{I I}

0,02

0,Q2 0,Q3 0,04

u I / ,

ni=1 1 1 1 IN 1

....

I I I 1/

i I I A I

I I AI 1 I

k.1"V i I I. I

1 1/1 I I I

1/11 1 I !

1 1 1 I 1 I

11 I 1 1

I I

I ^{1 I} I

I

I ^{1 1}

I I

I

i

I

I I I I

I

1

I

I I

1 11

11 I I I I

I

I

^{i I 11}

I

I ^{I !}

I

0.05 0,06 Dbubble 0,5 = 0.00378 + 0,0171 UG + 0,00102 L * + 0,0001758 hw

cap

I

I I I

I 1

I I

I I 1 I I

I I I

1

I

I

I

I I I

! _I

I

I

i

0,Q7

165

Fig. 8. Test eddy-diffusion coefficient vs. values calculated by the correlation valid for bubble caps

of Fig. 8 that the use of vah-e trays results in a higher eddy-diffusion coeffi- cient and a more intensive liquid mixing on the tray.

Although Eq. (6) can be used for design purpOSf'S_ it is advisable to restrict its application to the load limits specified in this study and for the types of yalve and tray described before.

h_ll. [mm]

m u [m/sec]

v [m/sec]

w

kmol dissolved matter x krnol solution ::; [m]

A. [sq.m.]

D [sq.m./sec]

EltW EOO

Legend gate height

slope of equilibrium curve linear gas (steam) velocity linear phase velocity

dimensionless locus co-ordinate concentration ofliquid phase

co-ordinate

flow cross-section area of liquid eddy-diffusion coefficient

~furphree tray efficiency l\furphree point efficiency

6 Periodica Polytechnica )L 18/2 - 3

166

F = u·

y

^Qa [m/secl! kg _V

J

cU.m.

GM [kmol/hour]

L [cu.m./sec: ou.m./hour]

L* [cu.m./mh]

LM [kmol/hour]

Pe

S [cu.m./sec]

SM [kmol/hour]

Zc [mm, m]

Z/ [mm, m]

Zw [mm, m]

Q [kg/cu.m.]

gM [kmol/ou.m.]

'i i.

be g G AI o

I!'

K. ,11OL.vAR

gas load factor molar gas flow rate liquid load

liquid load referred to the unit length of the gate molar liquid flow rate

Peclet number

rate of liquid weeping through the tray molar rate of liquid weeping through the tray height of clear liquid on the tray

distance between inlet and outlet gate average width of liquid flow on the tray density

molar' density marking

ratio of slopes of the equilibrium curve to the operating straight line

marking

Subscripts inlet

located at injecting network vapour phase

molar

located at inlet gate gare

Summary

For determining the numDer of trays required for tray distillation towers, tray efficiency must be known. There is a correlation between Murphree point efficiency and tray efficiency, depending on the material system and the extent of liquid mixing involved in the process.

For establishing the extent of liquid mixing, the eddy-diffusion coefficient must be known. Since there is some weeping in valye tray columns of a rate depending on tray design, the correlation proposed for static trays does not apply to valve trays. A correlation has been elaborated for calculating the eddv-diffusion coefficient. Evaluation of the test results was made on a model allowing for weeping. This correlation has proved to be suitable for determin- ing the effective number of trays required for tray distillation columns with a fair accuracy.

References

1. ~IoLl".-iR, K.: Periodica Polytechnica, ~I. E. 17, 4. 301 (1973)

2. TREYBAL, R. F.: ~Iass Transfer Operations, :'\ew York, ~IcGraw-Hill Book Company, 1968.

3. LEWIS, W. K.: Ind. and Eng. Chem. 28, 399 (1936)

4. "Tray Efficiencies in Distillf}tion Columns", Final Report, University of Delaware. :'\ew York, 1958.

5. "Bubble-Trav Design Manual". A LCh.E. Manual, :'Iew York, 1958.

6. K.AFAROY, V: V.: _Kz anyagatadas alapjai. (Bases of Mass Transfer. *) Miiszaki Konyv- kiad6, Budapest, 1967.

7. PERRY, J. H.: Chemical Engineers' Handbook. McGraw-Hill Book Company, Xew York, 1968.

8. BARKER, P. E.-SELF. M. F.: Chem. Eng, Sci. 17, 541 (1962)

9. Foss, A. S.-GERSTER. J. A.-PIGFORD, R. L.: A.LCh. E. Journal, 4,231 (1958) 10. MOLl",-iR, K.: Turbulent diffusion coefficient in yalve plate columns.* (Doctor Tecbn.

Thesis.) 1972.

11. :'IORMAl". W. S.-GROCOTT, G. J.: Trans. Inst. Chem. Engers. 39, 305 (1961)

Dr. Karoly M OLl'..\.R , H-1521 Budapest

* In Hungarian.