R&D NOTES Influence of Raiation on Prouct Yiels in a Film Boiling Reactor C. Thomas Aveisian, Wing Tsang, Terence Daviovits, an Jonah R. Allaben Sibley School of Mechanical an Aerospace Engineering, Cornell University, Ithaca, New York, NY 14853 DOI 1.12/aic.11388 Publishe online December 26, 27 in Wiley InterScience (www.interscience.wiley.com). Keywors: catalysis, reactor analysis, milti-phase flow, film boiling, boiling Introuction The film boiling reactor (FIBOR) 1 is analyze to show the influence of raiation across the vapor film on prouct yiels. The geometry is that of a horizontal, catalyst-coate, tube suspene in a pool of saturate methanol at atmospheric pressure. The vapor film surrouning the heate tube is the reacting volume an surface reactions are treate with an Arrhenius form of the reaction rate. Previous analysis on film boiling without catalytic reaction on a horizontal tube showe the influence of surface emission. 2 5 When couple with catalytic ecomposition, film boiling analysis has neglecte raiative effects. 1,6 In this note we set limits to this assumption. We begin by outlining the moel for film boiling with chemical reaction an present results for a horizontal tube with wall temperatures ranging from the minimum film boiling temperature neee to support a vapor film (45 K for methanol) to 18 K though most catalysts will ecompose well before this upper limit an practical operational limits may be uner 1 K. The present calculations are esigne to show general trens at high temperatures where raiation may be important. The species iffusion process is erive from the previously presente analysis. 1 All of the assumptions of that analysis are aopte here with the aition of those specific to raiation as iscusse in the Analysis section. We first consier the case where raiation is entirely etermine by surface emission between the tube wall an liqui/vapor interface an neglect volumetric absorption an emission; we then consier the effects of volumetric absorption an emission in the vapor film. Corresponence concerning this article shoul be aresse to C. T. Aveisian at cta2@cornell.eu. Current aress of Wing Tsang: Physical an Chemical Properties Division, National Institute of Stanars an Technology, Gaithersburg, MD 2899. Ó 27 American Institute of Chemical Engineers Analysis Surface emission with negligible gaseous absorption an emission Raiation across the vapor film influences heat transfer through an energy balance on a control volume in the vapor film which neglects kinetic energy as Z Z qhv na ¼ q na (1) A where the heat flux is the sum of contributions from conuction an raiation, which can be expresse as q n ¼ k vdt sat ð1 þ GÞ (2) G is a measure of the importance of surface emission from the tube wall with G ¼ C r (3a) where C r ¼ erðt4 w T4 sat Þ (3b) k v DT sat For G 1 surface emission is negligible compare to conuction across the vapor film an the analysis reuces to that of Urban et al. 1 With Eqs. 2 an 3, Eq. 1 can be expresse as q v Z ufl þ c pv ðt T sat Þgz ¼ 2 A Z / k v DT sat ð1 þ GÞ / (4) AIChE Journal February 28 Vol. 54, No. 2 575
The mass weighte average velocity in the / irection (u) an temperature istribution across the vapor film (T) are base on the conition that Pr Re (/v) 2 1. This limit leas to simplifie momentum an energy equations, with corresponing velocity an temperature profiles across the vapor film given by u ¼ gðq 1 q v Þ 2l v sin /ðz z 2 Þ (5) an a linear temperature istribution across the vapor film, z T ¼ T w DT sat (6) where z is the istance measure from the tube surface, DT sat 5 T w T sat an a no-slip bounary conition is assume at the liqui/vapor interface (the analysis neglects liqui motion an v 5 aty = as iscusse in Ref. 1). Substituting Eqs. 5 an 6 into 4 gives 3 Z sin / / 1 ¼ ð1 þ GÞ/ (7) 3B which is an integral equation. Differentiating Eq. 7 with respect to / yiels x / þ 4 3 x cot / ¼ 4Bcsc/ ð1 þ C rx 1=4 Þ (8) where x 5 4, G 5 C r x 1/4, an B is given by B ¼ 4c pv DT sat 2L þ c pv DT sat k v c pvq l v v q v (9) g q 1 q v q v when G 1 (no surface emission) Eq. 8, with the conition that / ¼ or equivalently (from Eq. 8) that, /¼ ¼ ; has an analytical solution 1,6 : lim /! x / R /! 1=4 ¼ ffiffi sin 1=3 ð/þ/ p 2 B 1=4 (1) sin 1=3 ð/þ Otherwise, the solution to Eq. 8 must be obtaine numerically using the conition for x at / 5 8 (where x 5 x o an ¼ ; from Eq. 8) that lim /! x / x o 3Bð1 þ C r xo 1=4 Þ¼ (11) A fourth orer Runge-Kutta numerical metho was use to solve Eq. 8 with x o given by Eq. 11. To test the accuracy of the numerical solution, Eq. 1 was compare with the numerical solution of Eq. 8. The results agree to less than 1% eviation over the ranges 45 K \ T w \ 18 K an \ / \ 1798. The particular reaction consiere is catalytic ecomposition of methanol with the overall reaction CH 3 OH? CO 1 2H 2 (i.e., two moles of hyrogen form at the expense of one mole of methanol). The rate constants for this reaction were taken from Ref. 6 base on measure heat transfer coefficients of a catalyst-coate tube in film boiling in methanol an plug flow reactor ata. It is important to note that in this moel we o not consier egraation of the catalyst (e.g., Pt-black) uring operation of the FIBOR which, nonetheless, can be an important consieration in practice. The hyrogen yiel is relate to the mass fraction of methanol at the tube surface as where 1 M /;H 2 ¼ - H2 2 Z / - H2 Ce RoTw E Y CH3OH;w/ (12a) (12b) E is the activation energy of the reaction, C is the frequency factor of the reaction, an Y CH3 OH,w is the mass fraction of methanol at the tube surface (z 5 ) which epens on /. The proceure for etermining Y CH3 OH,W in Eq. 12a when raiation is inclue is operationally ientical to when raiation is neglecte. 1 The analysis iffers from Ref. 1 by the presence of G 5 C r = in Eq. 8 an consieration of volumetric absorption in the vapor film. An integral metho is use to analyze transport of species in the vapor film with thir orer polynomials assume for the chemical species across the film. This approximation is the simplest approach consistent with satisfying bounary conitions at both the tube wall an liqui/vapor interface to give a reasonable functional form for the species istribution across the vapor film. The thir orer polynomial assumption satisfies four bounary conitions, two each at the soli/vapor an vapor/ liqui interfaces. Integral methos in which istributions (e.g., of concentration) are assume as in the case here may suppress errors in the compute yiels (Eq. 12a). Furthermore, the preictions are strongly epenent on the reaction rate appropriate for the catalyst uner consieration (e.g., Eq. 12b). Rate constants an parameters are formally inepenent properties of the system. However, the actual physical an chemical effects are a manifestation of the complex interaction between transport, raiation an chemistry. In that sense, the rate constants an parameters are tie to the moel. More etaile analyses for species iffusion may not be any more accurate than the rate constants use to preict prouct yiels. Absorbing an emitting gas The inclusion of volumetric absorption an emission in the vapor film complicates the analysis because the temperature istribution across the vapor film is then no longer given by the linear form of Eq. 6. A comparatively simple approach is taken which assumes the gas to be optically thin such that s 1 where s 5 j an j is the Planck mean absorption coefficient (m 21 ) which epens on temperature an gas composition. j can be calculate from the gas temperature T, an the mole fractions of CO an CH 3 OH as a molar average of the iniviual species absorption coefficients j i, 7 576 DOI 1.12/aic Publishe on behalf of the AIChE February 28 Vol. 54, No. 2 AIChE Journal
an Z ¼ z, h ¼ T T w, an h s ¼ T sat T w : D is a measure of the relative importance of volumetric absorption an emission to conuction: for large D gaseous absorption an emission are negligible, conuction across the film thickness ominates heat transfer, an a linear temperature istribution results (e.g., Eq. 6); as D ecreases, the temperature istribution becomes progressively nonlinear inicating that the gas absorbs an emits. Results Figure 1. (a) Variation of gas temperature across the vapor film for T w 5 5 K. Deviations from the conuction limit (linear variation) are observe for D \ 1. (b) Variation of gas temperature across the vapor film for T w 5 1 K. Deviations from the conuction limit (linear variation) are observe for D\1. Two parameters govern the importance of raiation: one for surface emission (G) an the other for volumetric absorption (D). We first consier volumetric absorption. Equation 14 is a nonlinear ifferential equation with no close form solution. However, Eq. 14 can be solve numerically by itself if D is consiere a parameter. Doing so allows etermining the conitions uner which volumetric absorption are important. That is, if the solution of Eq. 14 results in a linear temperature profile, conuction ominates an the analysis can neglect volumetric absorption; otherwise, it must be consiere. We first present the solution to Eq. 14 for a range of D then etermine which are most relevant to a FIBOR, using the example of methanol ecomposition consiere here. Figure 1 shows the numerical solution of Eq. 14 for \ Z \ 1 (i.e., \ z \ ) using a collocation metho. For D 5 1, the temperature istribution is linear which signifies that volumetric absorption is negligible. Slight eviations are observe for D 5 1 an significant ifference from a linear profile are seen for D.1 for both T w 5 5 K (Figure 1a) an 1 K (Figure 1b). Figure 2 shows the variation of D with parameters base on Eq. 15. We simply varie over the range in the figure to obtain the result shown. The numerical solution shows to be within the range epicte in the figure. For all combinations of conitions above the horizontal line (D 5 1) absorption is negligible. R 1 j ik e bk k j i ¼ rt 4 (13) The ata for j ik (T) 8 show that for CO an CH 3 OH a conservative (e.g., high) estimate for the gas mixture absorption coefficient ranges from 17 m 21 to about 26 m 21 at room temperature. Given that our results show that is uner about 2 mm (except for the singularity at / 5 188) the optically thin assumption is reasonable for almost the entire circumference of the tube surface. The temperature istribution across the vapor film in the optically thin limit 7 can be put in the following non-imensional form D 2 h Z 2 ¼ h4 1 2 h4 s 1 2 (14) where we efine D as D ¼ k 4rT 3 w j2 (15) Figure 2. Variation of D with T w at the inicate. For conitions above the blue line (D 5 1), volumetric absorption in the vapor film is negligible (cf, Figure 1). [Color figure can be viewe in the online issue, which is AIChE Journal February 28 Vol. 54, No. 2 Publishe on behalf of the AIChE DOI 1.12/aic 577
Figure 3. Scale epiction of vapor film thickness aroun a 5 mm iameter tube at the inicate value of T w. Divisions are in.5 mm an 22.58 increments. Re line inclues raiation; blue line neglects raiation. For 5 an 75 K cases, results are virtually coincient. [Color figure can be viewe in the online issue, which is Consiering the practical operational limits of a FIBOR, a lower boun of D using Eq. 14 is obtaine from the lowest value of k an highest (reasonable) values of T w, j an for a gas film comprise of methanol, CO an H 2. We estimate that k.3 W/mK (methanol 9 ), j 26m 21 using Eq. 13 an ata from, 8 \ 2 mm, an T w \ 18 K as an upper value limite by the melting temperature of the tube material (e.g., stainless steel). With these estimates we fin that D [ 5.7 which is the smallest value that is relevant to present conitions. Figure 1 shows that even for D 5 1 which is alreay too low base on the above estimate, the temperature istribution is nearly linear. As a result, for the parameter values consiere here there are no conitions where volumetric absorption woul be important when using methanol as the reactant liqui. Uner this circumstance, the solution of Eq. 14 is in effect given by Eq. 6 (as / 5 188 Y CH3 OH,w ecreases so that from Eq. 12a the contribution from the region aroun / 5 188 may also be small). Also, when volumetric absorption is negligible, the chemical reaction process itself has no influence on raiation because the absorption coefficient which is influence by the prouct species oes not then enter into the analysis. Consiering only surface emission from the tube with e 5 1 in Eq. 3b for illustration (e.g., formulations in the Surface Emission with Negligible Gaseous Absorption an Emission section), Figure 3 shows scale images of the structure of the vapor film aroun a 5 mm iameter tube at four surface temperatures. The re curve inclues surface emission an the blue curve neglects raiation). At low T w, the vapor film thickness is very close to the tube surface except for the singularity at / 5 188. Increasing T w also increases the vapor film thickness as a higher tube temperature enhances heat transfer which in turn increases the evaporation rate an thickens. At high temperatures, greater than about 1 K, the effect of raiation on the film thickness becomes noticeable, again increasing the film thickness compare to neglecting surface emission because of the ae contribution to heat transfer by raiation in aition to conuction. Figure 4 shows the variation of vapor film thickness with T w an tube iameter at a reference position of / 5 8( o ) for both the nonraiative an the raiative cases (i.e., from Eq. 11). When surface emission is inclue, o is larger than when raiation is neglecte for the range of tube iameters shown in Figure 4, as well as for T w above about 1 K. At lower temperatures, raiation has almost no effect on Figure 4. Variation of vapor film thickness at / 5 8 as a function of T w an showing the influence of raiation (re) vs. the surface emission neglecte case in blue. [Color figure can be viewe in the online issue, which is Figure 5. Influence of surface emission on vapor film thickness for three T w for a 5 mm iameter tube: otte lines inclue surface emission; soli lines neglect raiation. [Color figure can be viewe in the online issue, which is 578 DOI 1.12/aic Publishe on behalf of the AIChE February 28 Vol. 54, No. 2 AIChE Journal
Figure 6. Variation of G with T w at the inicate. For conitions below the blue line (G 5 1), raiative emission from tube surface may be neglecte. [Color figure can be viewe in the online issue, which is regarless of tube iameter. Figure 5 shows the variation of with / for T w 5 5, 1, an 15 K using a tube iameter of 5 mm as a reference. With increasing /, also increases (see also Figure 3). At 5 K, the ifference between the nonraiative film thickness an raiative thickness is negligible; at 15 K, the film thickness is 15% larger when surface emission is inclue. These effects are again attribute to increase heat transfer to the liqui/vapor interface by incluing raiation compare to neglecting raiation, an to the resulting effect of this increase on evaporation at the liqui/vapor interface. The parameter G (Eq. 3) etermines the influence of surface emission across the vapor film. Figure 6 shows how G epens on T w. As with Figure 2, we varie over the range shown in Figure 6 to illustrate its influence on G. The horizontal line correspons to conitions where surface emission an conuction exert a similar effect. For G 1, surface emission is important; for G 1 it is negligible. Precisely how large or small is require is etermine by the specifics of the solution. For example, comparing Figures 5 an 6 it is seen that at / 148 an 15 K, 6 lm an Figure 5 shows that there is a significant effect of raiation. From Figure 6 at these same conitions, G 4. Figure 7. Hyrogen yiel (per unit area of tube surface) as a function of T, (a) 5 3 mm, (b) 5 5 mm, (c) 51 mm, an () 5 15 mm. Re lines inclue raiation an blue lines neglect raiation. Insets show preicte yiels below T w 5 7 K. [Color figure can be viewe in the online issue, which is AIChE Journal February 28 Vol. 54, No. 2 Publishe on behalf of the AIChE DOI 1.12/aic 579
vapor film shoul be inclue in the analysis of film boiling with chemical reaction. For the conitions examine here, volumetric emission is not an important consieration an surface emission from the tube influences prouct yiels at wall temperatures only above about 1 K. Above this temperature, incluing surface emission in the analysis results in higher methanol mass fractions at the tube surface, larger vapor film thicknesses, an higher prouct yiels compare to neglecting raiation. Two nonimensional parameters are shown to etermine the importance of raiation, one concerning volumetric absorption an the other concerning surface emission. Figure 8. Variation of methanol mass fraction at tube surface for a tube iameter of 5 mm. Re curves inclue raiation an blue curves neglect raiation. [Color figure can be viewe in the online issue, which is Figure 7 shows the preicte hyrogen throughput as a function of T w an tube iameter. The hyrogen yiel was compute from Eq. 12 as outline in the surface emission with negligible gaseous absorption an emission section. Incluing raiation in the analysis increases M * 2/,H 2 (Eq. 12) for T w above 1 K but that there is virtually no effect at T w \ 7 K as inicate in the inset figures. Figure 8 shows the variation of Y CH3 OH,W with / at two temperatures (75 an 1 K) for a 5 mm iameter tube for illustration. The area uner the curves shown in Figure 8 is the integral in Eq. 12a. A slight increase is evient when raiation is inclue (re line) compare to when raiation is neglecte (blue line). This effect is ue to increase evaporation of methanol when raiation is inclue. The higher methanol wall mass fraction at T w 5 75 K compare to T w 5 1 K is the result of T w strongly influencing reaction rate (Eq. 12b) such that at low temperature more methanol accumulates at the surface without being converte compare to high temperature. While M * 2/,H 2 is proportional to the area uner curves like those shown in Figure 8, the higher area at lower T w is compensate by the much stronger increase of - H2 (Eq. 12b) as T w increases to prouce the increase of M * 2/,H 2 with T w shown in Figure 7. The effect of raiation on preicte hyrogen prouction is, as expecte, important only at relatively high temperatures here being greater than about 1 K for the conitions of the calculations as shown in Figure 7. In practice, the operation of a FIBOR woul be at temperatures which are high enough to rive prouct yiels to appreciable levels an to maintain film boiling yet without compromising the integrity of the tube material an catalytic coating. Concluing Remarks The operational range of parameters is ientifie where raiation effects either from surface emission from the tube wall or volumetric gaseous absorption an emission in the Acknowlegments The authors are please to acknowlege the support of this work by the National Science Founation uner grant no. CTS 5-15 with Dr. Patrick Phelan as the Program Director. The authors also thank Dr. Alfonso Ortega of Villanova University for his interest in this problem an conversations with Mr. Sungreyl Choi of Cornell. Notation A 5 area (m 2 ) c pv 5 specific Heat [J/(kg K)] 5 tube iameter (m) e bk 5 spectral blackboy emissive power (W/(m 2 -lm) g 5 acceleration ue to gravity, (m/s 2 ) h 5 enthalpy (J/kg) k v 5 mean vapor thermal conuctivity [W/(m K)] L 5 latent heat (J/kg) M* 5 mass throughput of component [kg/(h m)] n 5 normal vector Pr 5 Prantl number q 5 heat flux (W/m 2 ) T 5 temperature (K) T w 5 temperature at the tube wall (K) T sat 5 temperature at the vapor/liqui interface, DT sat 5 T w T sat (K) v 5 velocity vector [m/s] v 5 vapor velocity in y irection z 5 istance measure normal to tube surface (m) Greek letters 5 vapor film thickness (m) q 5 ensity (kg/m 3 ) e 5 emissivity, imensionless k 5 wavelength (m) l 5 viscosity [kg/(m s)] r 5 Stefan-Boltzmann constant [W/(m 2 K 4 )] / 5 angle measure from bottom of tube (ra) v 5 arc length aroun tube circumference (/2 /) Subscripts v 5 vapor l 5 liqui Literature Cite 1. Urban BJ, Aveisian CT, Tsang W. Film boiling with chemical reaction: analysis of an alternative metho for hyrogen prouction. AIChE J. 26;52:2582 2595. 2. Sakurai A, Shiotsu M, Hata K. A general correlation for pool film boiling heat transfer from a horizontal cyliner to subcoole liqui. I. A theoretical pool film boiling heat transfer moel incluing raiation contributions an its analytical solution. J Heat Tran. 199;112:43 44. 58 DOI 1.12/aic Publishe on behalf of the AIChE February 28 Vol. 54, No. 2 AIChE Journal
3. Liu MH, Yang YM, Maa JR. A general correlation for pool film boiling heat transfer from a horizontal cyliner to saturate binary liqui mixtures. Int J Heat Mass Tran. 1998;41:2321 2334. 4. Nishikawa K, Ito T. Two-phase bounary-layer treatment of free-convection film boiling. Int J Heat Mass Tran. 1966;9:13 115. 5. Sarma PK, Subrahmanyam T, Rao VD, Bergles AE. Turbulent film boiling on a horizontal cyliner. Int J Heat Mass Tran. 21;44:27 214. 6. Okuyama K, Iia Y. Film-boiling heat transfer with a catalytic ecomposition reaction. JSME Int J Ser B. 1994;37:123 131. 7. Sparrow EM, Cess RD. Raiation Heat Transfer, Revise Eition. Belmont, CA: Brooks/Cole Publishing Co. 197:2, 215, 25. 8. Butler RAH, Engler DL, Armstrong JC. The virtual planetary laboratory molecular spectroscopy atabase. 27. Available at: http://vpl. ipac.caltech.eu/spectra/. 9. Vargaftik NB. Hanbook of Physical Properties of Liquis an Gases. New York: Hemisphere Publishing, 1975:44 46, Manuscript receive May 2, 27, an revision receive Oct. 2, 27. AIChE Journal February 28 Vol. 54, No. 2 Publishe on behalf of the AIChE DOI 1.12/aic 581