EXPERIMENTAL TECHNIQUES
Extension of Standard Methods for Determining Diffusivities of Evaporation Solvents James M. Pommersheim Bztckuell Criiuersitg, Lewisburg, Pa. 17837
A model has been developed which governs the quasi-steady-state stagnant film diffusion of a solvent into an inert gas. Both open and closed systems with varying transport area have been treated. The model was tested experimentally against data from a closed system and gave excellent agreement.
The quahi-steady-state model for the stagnant, film eva1)orat,ioii of a solvent' has beeii widely used (hltshuller ant1 Coheii, 19GO; Larseii, 1964; Lee and Wilke, 1954) as the I m i s for the experimctitul determiiintioii of vapor diffusivities. Interest has liceii cciiterecl iiiore oil traiiq)ort rates tliaii 011 interface levels. III i)articular, tlie Stefan or - h i o l d diffusioii cell (Arnold, 1944) and various modification* (AIcIielvey and IIoelvlier, 1957) have been used for tliffusivit'y deteriniiiatioil,.. 111 all iuch cells, tlie transl)ort xrea is coiiht:iiit : i i d the diffusioii i. iixiret:irdetl, siiice the solveiit vapor i. sn-elit away a t the top of the cell 1)y:iii inert gas. Lit'tle :ittciitioii ha> beeii focuwl oii the cases of vat,i:il)le traiis1)ort w e a or uiiret:ii.ded diffusion, which c a n he of prnctical iiiiportaiic~c.The ~)i,eseiit study tlcvelops a geiierul mathein:itic~:il modcl which eiicornl) e\ tliesc care CIRCULATION
I I
(7) Eyiiatioii T hah bee11 derived (13eiinett :uid Myers, 1960; R c l t y et nl., 1069), a n d its tlifferenti:tl form is used as t'he Insi.: for espe~iinental piwiictioii of solveil t' diffiisivities iii a Stcfnii or .\ri)old cell. The espciirucnt a1 li~nitat~ions of such cells are tlisciiq.;rd by Alt~*hiillerand C'nhen (1960). Closed System. IVheiiever 1' is totally closed, evaporatioii will proceed tonartl the point whcrc vapor saturat,es t h e whole system, while system pressiire rises toward the sum of t h e initial total i w s s u r e ant1 vapor pressure. Since system total prwsiireq ( ~ 3 1 1never doul)le, aiid evaporation proceeds weir more slon-ly in this caw, the effects of pressure ctiffiiiion can l x ueglrcted. Thus, ~ i t hretarded evaporation, diffusioii \vi11 \)e relatively rapid outside t.lie well, so th:it, at aiiy fixed time only one partial pressure of A will esist ill the gas q)ncv. IYitli 110 .I iiiitixlly present' iii the gas .;pncc, (Y c a n be writtcii a s
-&I
'
IWELL I L - - - - -
I I I I I I
I I
7 - - - - - - 1
A Figure 1 . General system for quasi-steady-state stagnant film diffusion
Experimental
'The iiicreav in total gas voluniie c:iu.wl by evaporation ( a n i i lw iirglrctetl, +iiicc for i i i o r t liquid.: pa.; v r ~ l u n i eesceed ~ coi~rc.lmiidiiig liqiiitl voliiines niore thmi a lrinidictlfoltl. TIowover, if iiitcwht i h rentrrctl u j ~ o i ithe soliitioii a t loiig times, where the \vliolc terii :iiq)roac~hest lie hoturntion presciit. a t the intcrfnce, the variation in Ti must be considered. Thuq, ne:ir satui,ntioii the value of 1' needs t o be kno\vn accuraDely, siiire the argmiieiit' of the logarithm in Equation 8 approaches uuitg. In t#liiscase CY heconies I
I' 2
\
Partially Closed System. In some situations the system may be only 1)artially closcd, in the s e n ~ ethat the pressure is constant even though diffusion is .still retarded. V, then, noiild be only partially clowtl. For (,xaniplc, a n open drum of volatile solvvnt stored iu a closed rooni m a y alloiv i m m u r e equilihratioii with the atmoqphcre, h i t not at the eslmise of al)preciable solvent locs from the rooni, at, least within sonie initial period. I n s u c h a ctnsc with P ( z ) constaiil, 1Squatioii 6 siinl)lifics to
An esperinientmalclosed system study, was I methanol-air as t,he solvent-inert gas pair. The sign resembled that' shown in Figure 1 with 6' totally closed arid S(zJ constant. The diffusion cell itself consisted of a constant-diameter buret, sulyorted on a flat glass plate. Internal pressures were measured using a modified mercury barometer, calibrated t o sense pressure changes b y only part'ially evacuat'ing the air space above the mercury column. Aillowance was made for head change caused by air space compression. An inverted 30-liter bell jar sealed with soft wax, insoluble in methanol, closed the system. The relative dimensions of the bell jar and buret were such that the increase in volume arising from evaporation could be neglected except a t the very longest' times. The entire system was thermostated (10.4OC) in an air-cooled room in which internal circulation was continuously maintained. The system was covered except, when interface levels were determined. In making a run, t h e diffusion cell was first filled with solvent t o a known level, and brought into the rooni. Then, with air circulation maintained, the cell was opened and the bell jar was placed over the rvell. The tinier was started and the initial interface level and initial pressure (barometric,) were recorded. Then the bell jar was sealed. System temperature and internal pressure \rere periodically recorded. Interface levels were measured with a cathetometer. The diffusion path length \vas always taken t o be the distance from the bottom of the liquid meniscus to the top of the column, no correction being made for meni;wus curvature associated with surface teiision or eddies associated with mixing a t the column top (Lee and Kilke, 1954), or cell end resistance associated with molecular diffusion external to the cell (hndrew, 1955).
For all but the 10iigeht~times, the anioiint of el-uporating solvent corresponded Lrith t,he increase in internal pressure, within esl)erimental error. This indicated that, uiicleaired solvelit sorlition onto the glahb or \vas surfaces vias not. taking 1)lare. The filial int,er~ial1)ressure reniaiiied co~istnlit,iiidiwtinp that, the seal was effectil-e. the tern reriiaiiiiiig tot:illy ~ 1 0 4 -\t . very loiig tiinch, methanol coildeliset1 oii the i n 4 c wrfnce of thc 1)cll jar: I)rit ag:iiri with I I O caliaiige i ~ itcrt:il 1>resqlllc. Ind. Eng. Chem. Fundam., Vol.
IO, No. 1 , 1971
155
_.
~
RETARDED ___ UNRETARDED - CONDENSATION - - - - -
/
'
I
IO2 N
I
0 N
.I
2 101
SOLVENT ' METHANOL INERT . AIR
I
Po= I ATM To=25
'C
I
lo0
I
lo4
IO6
lo5
Io7
t , SEC.
Figure 2. Retarded quasi-steady-state stagnant diffusion in closed system of constant transport area
film
Results and Discussion
With a clobed iy-tem of coli-taiit cro+ Yectioii, Equatioii~5 and 8 iwluce to
For tlii. ~1111eii~e~ the i1itegr:il iii 1~;qii:itioii 11 niust IIC illt.cgi,:itetl nunieric:illy. 'I'he>e equation- were tested usiiig the ~olveiit-iiiri~tgas 1):iir methanol-:iir :it, 25'C : i i i d 0.99a t m initial pre.$iire. The diffu,*ioii coefficient used for this syqteni (0,1530 cm2 per second at 1 atni : ~ n d2 5 O C ' ) was c:ilcdatcd ii\ing the Hirschfeldcr, I%ii.d,aiid Sl~otz(1949) Iiietliod :I* modified by C'hen and Ot'hmer [1962). This coefficient lies c l o w t o the experiiiieiitnl v:diie of 0.1620 c m * per second rel)orted by Reid and Sherwood (19581, thnii th:xt. calculated directly u*iiig the Hirschfeldcr equatioii. Figure 2 coiii1)arcs the theoretical curve detei,~iiiiiedfi,om Equation. 11 and 12 and the experimeiit:il d:ita. Siiice t'lie diffusivity is ~)i~cdeteriiiiiiet1 from t'heory, t'lie theoretical bolutioii coiit:iiii.: 110 :icljn~tal)lecoii,+t:ints,:iii(i; for iiio+t times, the model aiid the d:it:i are in cscclleiit agwemeiit. The deshetl line cori~xpoiidsto the open system or uiiree (Ilquation 5 ) . The solid line is the asymptote for retarded diffu,+ion, correspo~idingto that value of t which makes the ai~giimentof the logarithm in Equation 12 equal to unity. Since cvajtoratioii of methanol was fairly slow at' 25' C y , the es~~eiiiiientnl data followed the uiiretartled path for iie:i~,lyn day before the diffusion begaii to become retarded. The ngrcement between the theoretical curve and the ex~)eriiiieiit:iltlnt:i Iioiiit. in the retiuded regioii :iI)pe:irs t o he much 1)etter t1i:iii iii the uiiretarded 1,egioii. However. hilice thc cooidin:itti of Figrii,c 2 are logai,ithniic, clevintioiis :it > l i ~ i . ttime. ~ r :ijq)enr mngiiified. 1niti:il uii.tently effect. :iisoci:ited with the buildup of the st:igii:iiit film witliiii the crll itqelf were c:ilcul:itctl to lie iic~g1cc~t:il)lc. l7-iiig tlic method of Lee niid JVilke, it \viis wtini:ited that tlie ccll reuclietl 99,9yo of steady htate iii 1~s.: than :I niiiiiitr. Soiiic coi~rcctii~ii f u i , diffii.ion extei,nal to the cell 156 Ind. Eng. Chem. Fundom., Vol. 10, No. 1 , 1971
may he needed, however. Thus, since 110 forced turbulence was maintained nit'hin the system, it must he assumed that a t any time only one coiiceiitratioii of solvent exists outside the cell. This is likely t80be untrue at' shorter times, although for suitably slow evaporating systems it would remain valid iidreiv (1955) has pointed out that t.hi.: end reses the effective diffusion leiigth by d o , % nt most. Thiq would cause diffusioii t o be somewhat slower than expected. Forced circulation of the contents of t'he would eliminate this effect,, but partly a t the expen somewhat shortened diffusional path length, c a u w i by turbulence a t the top of the cell (-iltshuller and ('oheii, 1960). For the same system, an air bath would not mitigate temperature gradients caused by evalloration or rever,se thermal diffusion nearly as much as a water bath, aiid iniivanted coiivection curreiit'ij may be set up which increase vapor traiisport, especially a t short times n.here evaporation rntes are higher. However, even for uiiretarded diffusion of more volatile vapors than the present system, such temper:iture gradients are always less than 0 . 2 O C ' (Lee mid Wilke, 1954), so that such effects are probably small. Figure 2 shoivs short-tinic diffuiion i,ate* sonicwhat' higher than exlmted. 'I'hi.: doc- not' imply that' interface cooling is resl)onail)le, or that end 1,esistance is ahelit', since ~ i d o n i erroi's may result iii the higher rates. Insufficieiit short'-time data ~ v e r ecollected to ertimate the end effcct cori,ectioti t o tlie nie:isurcd cell cleptli. However. sni:ill diffusional end effect, \\-oultl not affect tlie accuracy of the solution a t lo11gcr times. T h e iice of the reported eq)eiimentiil diffusion coefficieiit, 6y6 liigliei~than that predicted by the Cheii-Othmer correlation, \voultl briiig the niodel iiito almost exact ngrernieiit ivith t,he dat:i, iwolviiig t,lie probleni of the anornalou~lyhigh rat?. ol)servcd :it shorter tiniw;. However, interest, here remniiis ceiiteretl ujmn :I completely I)i,etlictive model, since, i ~ i g e ~ i c r dthe ~ diff'usivity \\-oultl iiot lie reported. (':~lculatio~is ~ v c 1 ' e:rlso nxidc using the Equation 9 aiialog for tlii,. syhteiii, \vhich :illo~vbfor the c1i:inges in system volume c:iuwi b y ev:ip(ir:itioii. Piiirc the total voliiiiic of liquid ev:ilw:itctl w:is lei. tlinii 0.05% of t#hevoliimc of the the hoi.izoiita1 asyiii1)totc of Figui,e 2 reinailis linear. Ilo\vcvcr, the diffcrcnce bct\veen the rnlculated diffusion tirile with v:ii,ial)le :ind lvith coii.taiit vulunie wa.: not negligible at' very Iotig tinie5.. l'he percentage devintioii 11ctn.cc1i these two value. i w s le+ than lYc :it 5 X l o 6secoiids, nncl less than 57c :it IO7 sworids, too mi:ill to be depicted 011 Figure 2 . ioiis usiiig the device are duo presented. TTe are not aware of a previous tlcrelopineiit of L: device which enaldes direct otiservat~ionof utihindered bubble iiiotioii for ail indefinite length of time without the use of ti traver3ilig movie camera. Ind. Eng. Chem. Fundam., Vol. 10, No. 1 , 1971
157
