Reference. R. K. Herz,

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1 Identifiation of CVD kinetis by the ethod of Koiyaa, et al. Coparison to 1D odel (2012) filenae: CVD_Koiyaa_1D_odel Koiyaa, et al. (1999) disussed ethods to identify the iportant steps in a CVD reation syste. In one ethod, they onsidered CVD on the internal of a ylindrial tube. They proposed that the liiting proess in a CVD reation syste ould be deterined by the pattern of plots of the log of the experiental groth rate vs. tube length for tubes of different length. In their Equations 1 and 2, Koiyaa, et al. presented a 1D odel of the syste. The onvetion of the reatant is represented by the ean veloity. For the diffusion liited ase, the reation is fast relative to the other proesses. The rate of diffusion of reatant to the an be represented by a ass transfer oeffiient that is appropriate for lainar flo. elo e develop equations for the 1D approxiation of the diffusion liited ase. diensionless ass transfer oeffiient, the Sherood nuber, is defined. The plot belo as produed by this 1D approxiation. The equations and Matlab progras are listed at the of this douent. The pattern shon in this plot agrees ith the diffusion liited plot in Koiyaa et al. This plot is also siilar to the pattern produed by the 2D odel that e developed in separate set of notes. Referene H. Koiyaa, Y. Shiogaki, and Y. Egashira, Cheial reation engineering in the design of CVD reators, Cheial Engineering Siene, vol. 54, no , pp , Jul

2 Introdution of the Sherood nuber to lainar flo reator Rihard K. Herz, Start ith the 2D equations shon previously for speies and using diensionless variables ith the pries dropped. 2 2 g = Da u + + x 2 x 2 + r 2 r r ( ReS) for speies (diensionless) 2 2 g = Da u + + x 2 x 2 + r 2 r r ( ReS) for speies (diensionless) No, neglet axial diffusion for large ReS and speify that there is no reation of at the. The produt ReS an be expressed as the ratio of onvetive ass transport rate to diffusive ass transport rate, and as the harateristi tie for diffusion aross the tube diaeter divided by the ean residene tie of fluid in the tube. See y notes on flo patterns in heial reators. lso, instead of desribing radial diffusion and reation at the exatly, e ill approxiate these obined proesses using a ass transfer oeffiient. This ill onvert the equations into ordinary differential equations. Sine onvetion and reation at the are both linear in the onentration, e an use the ean veloity, u, and ean onentrations,, to develop balanes around a diskshaped ontrol volue instead of our original ring-shaped ontrol volue. Using the original diensioned variables: g 0 = k u dx g 2 0 = k u + k dx R These equations are approxiations for gas reation in lainar flo (they desribe plug flo) and over-predit onversions by about 10. See y ourse notes on heial reation engineering, flo patterns in heial reators, for ReS >> 1 in lainar flo. The last ter on the right in the loer equation is the net rate at hih diffuses into the disk-shaped ontrol volue fro the. This rate is approxiated as being proportional to the differene beteen the and bulk-ean onentrations of, here the proportionality oeffiient, k, is alled the ass transfer oeffiient. The ass transfer oeffiient k has units (/s). The ter (2/R) is the area per unit volue of reator, here R is the radius of the reator.

3 Sine is being onsued at the, and <, this ter is negative, so it is onventional to reverse the onentration ters: g 2 0 = k u k dx R No e onvert bak to diensionless variables to get: 0 = Da u Sh dx here ' g ' ' ' ' ' 2 g g Da k / D / R ; a ' u u ' = 1 ; x ( 2 x/ R) /( ReS) ; u ' i i ;,0 ',0 and here the Sherood nuber is: Sh 2Rk D No e have a syste of to ODEs. Dropping the pries: Da g dx = g Da Sh( ) dx = Our only proble no is that e don't kno. Sine both ass transfer and this reation are first-order proesses, e an do soething about this. The flux of fro the bulk gas to the equals the rate at hih reats at the : k = k little algebra gives us: kk k ( ) = = k k + k 2Rk Sh D

4 Mass-transfer liited ase: For the liiting ase here the rate of reation at the is fast, k >> k, k k and >>. Sh = Sh = 2 Rk / D. In Koiyaa et al. this ase 3 is loosely alled "diffusion liited." The fil groth rate equals k ( /s ) (ol/ ). Sine Sh and, therfore, Sh does not vary ith R, k is proportional to (1/R). Surfae-reation liited ase: For the liiting ase here the rate of reation at the is relatively slo, k << k, k k and. Sh = 2 Rk / D. Note that, hereas, > suh that k ( ) > 0. Note that Sh for this ase is proportional to R. lso note that the value of uh less than the value of 3 fil groth rate equals not vary ith R. No our diensionless equations are: dx = Da g g Da Sh dx = k is indepent of R suh that Sh for the ass-transfer liited ase, sine Sh for this ase is k k <<. The k /s (ol/ ). The surfae reation rate onstant k does The initial onditions are, at x = 0, diensionless = 1, diensionless = 0. The equations are in the for of those for a standard series reation syste. The equation for an be integrated easily by separating variables, and then the solution for as a funtion of x an be substituted in the equation for. This equation an be integrated using the integrating fator approah. Gas-phase reation liited ase: Sh for the ass-transfer liited ase: g Da << Sh. The values of the Sherood nuber an be deterined fro analytial solution of the equations for lainar flo and radial diffusion, or solution of the equivalent heat transfer proble to obtain the Nusselt nuber for heat transfer (the Sherood nuber is soeties alled the Nusselt nuber for ass transfer). For fully developed lainar flo entering a tube in hih ass transfer to the ours ith onstant onentration at the, e.g., fast reation suh that the onentration is approxiately zero, the Sherood nuber varies ith diensionless axial position x' in the folloing ay, as obtained fro the solution of the governing partial differential equations:

5 diensionless x': 0, 0.002, 0.02, 0.2, 0.4, Sh:, 12.8, 6.0, 3.71, 3.66, 3.66 here diensionless x' = (2x/R)/(ReS), Sh 2Rk D Sine Sh does not vary ith tube radius R, k is proportional to (1/R). [e.g., Kays and Craford, "Convetive heat and ass transfer," 2nd ed., MGra-Hill (1980), p. 111, for analogous heat transfer ase]

6 1D SYSTEM - Matlab listing When you opy to Matlab do not inlude page footers & ay need to add return beteen odel CVD reations in tube (g)->(g) (g) -> fil at (s) this progra requires the folloing -files: vd_1d_derivs. vd_1d_sherood. THIS PROGRM CURRENTLY IS FOR MSS-TRNSFER LIMITED CSE sine funtion Sherood, hih is used by funtion vd_1d_derivs, supplies only Sh for ass-transfer liited ase arning('off') otherise get arnings fro taking log for plot share global variable values ith ain and funtions global Da Rratio GTL INPUT VLUE - GTL IS 1 for Gas Transport Liited Case other for Surfae Reation Liited Case GTL = 1; Rratio = ratio of tube radius R to a referene radius for Rratio = [ ] Dakoehler nuber for -> rxn in gas Da for gas rxn prop to R^2 Da = 200Rratio^2; initial onditions of depent integration variables of ODEs diensionless inlet onentrations 0 = 1; 0 = 0; 0 = [0 0]; speify span in indepent integration variable of ODEs diensioned tube length relative to a referene tube length xd0 = 0; xdl = 2; xdspan = [xd0 xdl]; for onstant-length tube, diensionless x prop. to 1/R^2 xspan = xdspan/rratio^2; span in diensionless tube length integrate using standard atlab funtion ode45 [x ] = ode45('vd_1d_derivs',xspan,0); = (:,1); = (:,2); [ro ol] = size(); f = (ro); X = (0 - f)/0; f = (ro); onvert fro diensionless x to diensioned xd relative to referene tube length xd = xrratio^2; if GTL == 1 alulate fil deposition rate (ol/2/s) = k(/s)(ol/3) k = ShD/(2R), so k and fil deposition rate prop to /R frate = /Rratio; else for surfae reation liited, rate = k(/s)(ol/3) k is indepent of R frate = ; probably a better ay in atlab than this to have different olor urves for eah ase on plot but this oes to ind if Rratio == 0.5 RateMin = frate; xdmin = xd;

7 Min = ; Min = ; elseif Rratio == 2.0 RateMax = frate; xdmax = xd; Max = ; Max = ; else RateMid = frate; xdmid = xd; Mid = ; Mid = ; plot rate and distane don tube as diensioned values relative to referene diensioned values of rate and tube length plot(xdmin,log10(ratemin),'g',xdmid,log10(ratemid),'r',xdmax,log10(ratemax),'b') title('green = 0.5 Rref, red = Rref, blue = 2 Rref, Rref = referene tube radius') ylabel('log10(fil deposition rate) (arbitrary diensional units)') xlabel('distane don tube (arbitrary diensional units)') axis([0 xdl ]) TKE LOOK T GS PHSE CONC figure(2), plot(xdmin,min,'g',xdmid,mid,'r',xdmax,max,'b') title(': green = Rref/2, red = Rref, blue = 2Rref, Rref = referene tube radius') ylabel('diensionless ') xlabel('distane don tube (arbitrary diensional units)') figure(3), plot(xdmin,min,'g',xdmid,mid,'r',xdmax,max,'b') title(': green = Rref/2, red = Rref, blue = 2Rref, Rref = referene tube radius') ylabel('diensionless ') xlabel('distane don tube (arbitrary diensional units)') LISTING OF FUNCTION FILE vd_1d_derivs. funtion dx = vd_1d_derivs(x,) funtion vd_1d_derivs uses funtion vd_1d_sherood share global variable values ith ain and funtions global Da Rratio GTL = (1); = (2); if GTL == 1 for gas-transport liited ase, k >> k so Sh = 2Rk/D and k is proportional to (1/R), so Sh indep of R here take into aount variation of Sherood nuber in entrane length - not uh hange fro using onstant 3.66 Sh = vd_1d_sherood(x); else for surfae-reation liited, k << k so Sh = 2Rk/D and Sh is proportional to R ShRef = 0.3; arbitrary value but ust be << 3.66 Sh = ShRefRratio; dx = -Da; dx = Da - Sh; dx = [dx; dx]; return as olun vetor

8 LISTING OF FUNCTION FILE vd_1d_sherood. funtion Sh = vd_1d_sherood(x) this funtion supplies the Sherood nuber for ass-transfer to ylindrial tube for fully developed lainar flo and onstant onentration, e.g., a onentration near zero for fast reation variation of Sh in entrane region has sall effet vs. using liiting value of 3.66 everyhere for the values of inputs in ain progra vd_1d aount for variation of Sh in entrane region here onentration profiles are developing fit urve to Sh vs. diensionless x in entrane region oeffiients of 3rd order polynoial fit of log(x) vs. log(sh) see belo oef = [ ]; if (x<0.001) Sh = infinity at x = 0 but don't bother Sh = 12.8; elseif x < 0.2 use fit in entrane region lnsh = polyval(oef,log(x)); Sh = exp(lnsh); else past entrane region, use liiting Sh Sh = 3.66; fit Sh in entrane region Sh = [ ]; x = [ ]; lnsh = log(sh); atlab log = ln lnx = log(x); oef = polyfit(lnx,lnsh,3); 3rd order polynoial hek urve at ore points than ere fit to see if have deviations beteen fit points lnxp = log(0.001):0.1:log(0.3); lnshp = polyval(oef,lnxp); plot(lnx,lnsh,'b',lnxp,lnshp,'k') title('blue = data, blk = fit') E CREFUL ith 2 fators If hange diensionless x' = (2x/R)/(ReS) to get rid of 2's in ODEs, then have to re-fit the funtion in Sherood.!!!! Sherood. diensionless x' = (x/r)/(res)