Modeling Local and Advective Diffusion of Fuel Vapors to Understand Aqueous Foams in Fighting Fires. Final Report

Transcription

1 Modeling Local and Advective Diffuion of Fuel Vapor to Undertand Aqueou Foam in Fighting Fire Final Report Author: Andrew Brandon Advior: Dr. Ramagopal Ananth Naval Reearch Laboratory, Wahington, D.C. Abtract The purpoe of thi project wa to invetigate fuel vapor uppreion by aqueou film and foam layer. Immediately after application, aqueou film and foam layer have the effect of uppreing fuel evaporation, however thi uppreion i not contant over time. If enough time elape, the fuel vapor concentration above the film and foam layer can reach level characteritic of an uncovered fuel pool. Thi can allow for reignition of the original fire. Experiment done by Leonard and William have tudied thi phenomena. It i unknown how the fuel move through the film or foam layer and the purpoe of thi project wa to model that movement by auming that the fuel diolve and diffue through the aqueou layer. In addition to delivering a model that can imulate thee experiment, I promied that thi model would be capable of finding an effective diffuion contant D F for the foam and film. Thi would be done by minimizing the reidual between the experimental and numerical reult. In order to enure that the model wa correctly imulating the experiment, the model wa extenively validated. After validation, the model wa applied to tet cae and film layer data collected by Leonard. After analyzing the reult of imulating Leonard experiment, it appear that the diolution and diffuion tranport mechanim i not the primary tranport mechanim of fuel through the film layer.

2 1 Background To combat fuel pool fire, the United State Navy employ everal type of fluorinated fire fighting foam. Once applied to a pool fire, a portion of the liquid in the foam drain over a hort period of time, depoiting a layer of film on top of the fuel pool. The fluorine urfactant in the foam i reponible for decreaing the urface tenion of the foam olution, allowing the aqueou layer to float on the urface of the fuel pool. Thi film uppree further evaporation of the fuel. However, everal experiment have hown that thi uppreion of fuel vapor i not contant over time. In the 1970, Dr. Joeph Leonard et al at the Naval Reearch Laboratory (NRL) in Wahington, D.C. began teting the uppreion ability of the then new aqueou film forming foam (AFFF) [1]. To tet AFFF, they took the film that the foam create and placed a pecified amount on a fuel pool. Then they meaured the concentration of fuel vapor over a pecified amount of time. Their reult confirmed that the film created by AFFF wa capable of initially uppreing the vapor of everal fuel type [1]. What they did not expect to find wa that after the initial uppreion, the concentration of fuel vapor increaed with time. In ome cae, fuel vapor concentration reached level characteritic of when foam i abent. Recently, thi phenomenon ha gained interet. Dr. Bradley William, alo of NRL, i currently working on an experiment imilar to Dr. Leonard. Hi experiment involve placing an aqueou foam layer on the urface of a fuel pool rather than the film layer. When meauring fuel vapor concentration level over a period of time, Dr. William ha alo found that the concentration of fuel vapor increae with time after the initial uppreion by the foam layer. Figure 1 i a depiction of the application of an aqueou foam and the reulting film and foam layer. Figure 1: Application of an aqueou foam with foam and film layer formation on a fuel pool [2]. 1

3 The exact mechanim of fuel tranport through foam or film layer are not known. However, there are currently two main hypothee. The firt i that the fuel vapor are able to diolve in the aqueou olution and then diffue throughout the olution. The econd theory i that the fuel i able to emulify into the aqueou olution and then i tranported to the urface due to denity difference. How fuel i tranported and by which of thee two mechanim were the driving quetion behind thi project. To anwer thi quetion, I modeled the experiment of Leonard and William by imulating either a film or foam layer on the urface of the fuel pool. The cae of a foam layer on the urface of a film layer wa not conidered. To model the tranport of fuel vapor through the layer, I aumed that the tranport mechanim wa that of diolution and diffuion. Thi diffuion would be governed by an effective diffuion coefficient D F and D F wa found by minimizing the reidual between the experimental and numerical reult. Once the value of D F capable of reproducing the experimental data wa found, it would be analyzed to determine if our tranport mechanim aumption wa correct. 2 The Model The experiment of Leonard and William involved a cylindrical container that wa 20cm in height and 2.5cm in radiu. A fuel pool wa placed in the bottom of the container and then a film or foam layer wa placed on the urface of the fuel pool o that the fuel pool wa completely and uniformly covered. Above the film or foam layer i a fritted gla dik that wa attached to a pipe that run out the top of the container. Pure nitrogen wa then blown through the pipe and fritted gla dik at a contant velocity. The flow of nitrogen then move acro the film or foam layer urface, entraining any fuel vapor. The flow would then move through an outlet at the top of the container to be analyzed. See Figure 2 for a depiction of the experimental domain. Since the experimental domain i cylindrical and axiymmetric, the model domain will be the region outlined in red in Figure 2. The two domain that were modeled were the film or foam layer and the ga domain within the red box. Thee two domain are labeled Domain 1 and Domain 2 within the figure and the fuel pool wa repreented a the bottom boundary condition in Domain 2. The flow rate that were ued within the experiment were choen uch that the film or foam layer wa unditurbed o the film and foam layer were aumed to be tationary in the model. Alo, ince the concentration of fuel vapor in Domain 1 and 2 i mall, the change in denity in the ga of the container will alo be mall. Thu, the denity of the container will be aumed to be homogenou and contant, which leave u with an incompreible ytem. The third aumption in the model i that the fuel vapor will be tranported through the film or foam layer by diolution and diffuion. In order to model thee experiment, the axial and radial velocitie of Domain 1 2

4 Figure 2: A lice of the experimental domain from the top of the container to the bottom. and the concentration of fuel vapor in Domain 1 and 2 mut be found. Thu, the pecie fraction equation (1) and the Navier-Stoke equation (2) will be olved. In equation (1) and (2), Y i the mole fraction t + (vy ) = D 2 (Y ) (1) v 1 + v (v) = t ρ (P ) + η 2 (v) (2) or ma fraction (depending on the bottom boundary condition) of fuel vapor and v (u, w) where u i the radial velocity and w i the axial velocity [4]. Note that the gravity term in the Navier-Stoke equation ha been removed. Gravity ha been removed becaue there are no denity perturbation in the ytem and gravity erve to create flow due to denity difference. Now, after applying the aumption of incompreibility ( v = 0) and applying the differential operator in axiymmetric cylindrical coordinate, equation (1) and (2) tranform into 3

5 t + u r + w ( 2 z = D Y r r r + 2 Y ) z 2 u t + u u r + w u w t + u w r + w w z = 1 ρ z = 1 ρ P ( 2 r + η u r u r r + 2 u ) z 2 P ( 2 z + η w r w r r + 2 w ) z 2 (3) (4) (5) To olve equation (3) for the concentration (mole or ma fraction) Y, it i neceary to know D. There will be two value for D, one value for Domain 1, D A, and another for Domain 2, D F. The value for D A repreent the diffuivity of fuel vapor in N 2 and can be etimated uing kinetic theory. The econd value D F repreent the diffuivity of fuel vapor in the film or foam and thi value i unknown. Thi value, D F, wa found by comparing the numerical model to experimental data. In order to olve equation (4) and (5) for u and w, it i neceary to olve for preure, P. Solving for preure i extremely computationally expenive and determining boundary condition for preure can be ambiguou. Thu, to avoid having to olve for preure, the tream function and vorticity tranformation were ued on equation (4) and (5). The ubtitution required for thi tranformation are ψ ψ r u = 1 r z, w = 1 u r, and Ω = z w r where ψ i the tream function and Ω i vorticity [5,7]. Upon inerting thee ubtitution, equation (4) and (5) are tranformed into the tream function and vorticity equation. Note that in thee tranformed equation there i no preure term, P. Ω t + u Ω r + w Ω z = uω r Ω = 1 ψ r 2 r 1 2 ψ r r ψ r z 2 ( 2 + η Ω r Ω r r + 2 Ω z 2 Ω ) r 2 Thi tranformation leave u with having to olve equation (6), (7), and (8) in Domain 1, where u = 1 ψ r z, w = 1 ψ r r, and D A i the diffuion coefficient for fuel vapor in pure nitrogen N 2. Then in Domain 2, only equation (9) need to be olved, where D F i defined a the effective diffuion coefficient for fuel vapor in the film or foam layer. Equation (9) ha no convective term becaue the film or foam layer i aumed to be tationary, which lead to u = w = 0 in Domain 2. t + u r + w ( 2 z = D Y A r r r + 2 Y ) z 2 Ω = 1 ψ r 2 r 1 2 ψ r r ψ r z 2 (7) 4 (6)

6 Ω t + u Ω r + w Ω z = uω r ( 2 + η Ω r Ω r r + 2 Ω z 2 Ω ) r 2 ( 2 t = D Y F r r r + 2 Y ) z 2 (8) (9) 3 Solution Algorithm The olution for equation (6), (7), (8), and (9) were found uing a erie of finite difference algorithm. To olve (6), (8), and (9), an upwind differencing algorithm for convective diffuive equation preented in [6] wa implemented. Thi algorithm call for backward differencing to be applied to the convective term and centered differencing to be applied to the diffuive term. Thi algorithm ha a retraint on the timetep, t, that could be ued to dicretize the time derivative. Thi retraint i t < 4D τ 1 + max(u) r + max(w) z where max(u) and max(w) are defined to be the maximum radial and axial velocitie in the model domain repectively and τ min( r 2, z 2 ) [6]. The model wa deigned uch that the equation in both Domain 1 and 2 were olved uing the ame timetep value, t, but becaue there wa a different D and z value in Domain 1 and 2, the timetep had to be deigned uch that the upwind differencing cheme wa table in both domain. Thu, the D and z value that were choen to find t were the D and z value that reulted in the mallet timetep. Thi enured that the upwind differencing cheme wa table in both domain. Equation (7) involve no time derivative o it could not be olved in a manner imilar to the other equation. To olve equation (7), a ucceive over-relaxation, SOR, algorithm wa implemented. The SOR algorithm i an iterative olver and i imilar to the Gau-Seidel method except that SOR ue a relaxation parameter to achieve fater convergence. To find the optimal relaxation parameter, Chebyhev acceleration wa implemented in tandem with the SOR algorithm. Chebyhev acceleration recalculate the relaxation parameter at each iteration of the SOR algorithm uch that it i the optimal relaxation parameter for that iteration. Thi guarantee that the SOR olver converge a rapidly a poible. The ucceive over-relaxation with Chebyhev acceleration algorithm that wa ued in the model wa the algorithm preented in [8]. Centered differencing wa ued to dicretize (7) and the olution wa iterated upon following the iteration formula in [8]. Once the reidual of the olution field fell beneath a precribed tolerance, the olution wa preumed to have been found. For further information on the implementation of the SOR with Chebyhev acceleration algorithm within thi model, pleae ee the December 2011 progre report. 5

7 Once the olution algorithm had been choen, it remained to couple the olver. The algorithm for doing thi and finding the olution for u, w, ψ, and Ω at each timetep wa preented in [6]. Thi algorithm called for boundary condition on ψ to be derived from thoe on u and w and for the remaining boundary condition on u and w to be implemented within the code. Boundary condition for Ω are then determined numerically from the u and w olution field according to the vorticity definition Ω = u z w r. Normally, boundary condition on Ω would be derived from boundary condition on u and w, however thi algorithm call for the boundary condition on Ω to be found numerically. Letting the olution for u, w, ψ, and Ω at the n th timetep be u n, w n, ψ n, and Ω n, then the algorithm for finding u n+1, w n+1, ψ n+1, Ω n+1, and Y n+1 i a follow. 1. Determine the boundary condition on Ω n numerically uing the u n and w n field and Ω = u z w r. 2. Find Ω n+1 with the upwind differencing algorithm. 3. Find ψ n+1 uing SOR with Chebyhev acceleration. There i an Ω term in (7), however only the interior point of Ω n+1 are required to find ψ n Solve for u n+1 and w n+1 from ψ n Solve for Y n+1 uing u n+1, w n+1, and the upwind differencing algorithm. 4 Validation Simplified Domain Once the olution algorithm were implemented, it remained to check that they were done o correctly. Validation wa begun on a implified domain. Thi domain wa an axiymmetric cylindrical container like the experimental domain but without the pipe and fritted gla dik. Once the implified domain wa in place, work began on validating the pecie fraction olver (i.e. the routine reponible for olving equation (6) and (9)). The firt cae that wa teted wa a pure advection cae with only axial dependence. Thi correpond to the equation which ha the analytical olution of t + c z = 0 6

8 { 0 z Y (z, t) = c > t z 1 c t for an initial condition of Y (z, 0) = 0, a boundary condition of Y (0, t) = 1, and a uniform axial velocity c inide the container. To implement thi cae of pure advection within the code, the appropriate initial and boundary condition were implemented along with a mall diffuion coefficient. Thi had the effect of making the diffuion term mall without having to remove thoe term from the code. The diffuion coefficient that wa choen wa D = 10 4 cm2 and the flow peed c wa 4 cm. The reidual between the numerical and analytical olution field wa then taken and Figure 3 i a plot of that reidual field. Looking at Figure 3, one can ee a region where the reidual i fairly high. Thi region i the interface between Y = 0 and Y = 1. Within the analytical olution thi region i harp however finite differencing i unable to capture harp interface. Finite differencing will caue thi harp interface to become mooth and one can ee evidence of thi within Figure 3. Becaue the reidual goe to zero in all other region, the pecie fraction olver i working correctly for the pure advection cae. The econd cae that wa teted wa a pure diffuion cae with only axial dependence. Thi correpond to the equation which ha the analytical olution of Y (z, t) = 1 n=0 t = D 2 Y z 2 } 4 π(2n+1) π(2n + 1) e D( ) 2t 40 in ( π(2n + 1) for an initial condition of Y (z, 0) = 0 and boundary condition of Y (0, t) = 1 and z = 0 at z = 20. The implementation of pure diffuion within the code wa a much more exact tak and wa accomplihed by etting u = w = 0 cm. The diffuion coefficient that wa ued in the analytical and numerical olution wa D = 0.25 cm2. Figure 4 i a plot of the reidual between the analytical and numerical olution for D = 0.25 cm2 and w = 0 cm at time=1.5. Looking at Figure 4, one can ee that the olution are much more in agreement ince the reidual i on the order of Again, thi i due to the olution being mooth which allow finite differencing to capture it. Becaue the olver found the correct olution to the pure diffuion cae, in addition to the pure advection cae, one can conclude that the pecie fraction work correctly. Once the pecie fraction olver wa validated, the vorticity olver wa then teted. Comparing equation (6) and (8), one can ee that the equation are very imilar except for two term in equation (8). Thee two term require no dicretization, o to tet the 40 ) z 7

9 Figure 3: Reidual between numerical and analytical olution for pure advective flow for D = 10 4 cm2 and c = 4 cm at time=1.5. vorticity olver, thee two term were removed. In addition, the boundary condition in the pecie fraction olver were implemented in the vorticity olver and η wa et equal to D. Thi had the effect of olving for the pecie fraction manifeted a vorticity. Once thi wa done, a reidual between the two olution wa taken. If the vorticity olver wa correctly implemented, then the two olution would be imilar. In fact, the reidual between the two wa found to be machine accuracy. Thu, the vorticity olver i correctly dicretized and ha been validated. The lat olver that needed to be validated wa the tream-function olver. Thi olver i different in nature to the upwind differencing cheme o it could not be compared to the pecie fraction olver like the vorticity olver could. In the end, the tream function olver wa validated by comparing olution from the SOR algorithm to Matlab finite element olver. A erie of the boundary condition were implemented for both numerical olver and then the reidual wa taken between the two olution. The reidual wa conitently on the order of 10 4 o the dicretization and SOR algorithm were correctly implemented. With the three olver being validated, the experimental domain wa then 8

10 Figure 4: Reidual between numerical and analytical olution for pure advective flow for D = 10 4 cm2 and c = 4 cm at time=1.5. added. Once the olver were coupled, further validation began. Experimental Domain Deigning the experimental domain involved adding the pipe and fritted gla dik to Domain 1. Thi alo required the derivation of new boundary condition on ψ. Table 1 i a lit of the boundary condition correponding to the labeled boundarie in Figure 5. Several value in Table 1 need to be dicued before continuing. The value Y ur i the mole or ma fraction of fuel vapor in air due to the vapor preure of the fuel. The value of α i c ince the width of the fritted gla dik i 2.25cm and the value of c i the input flow peed. The boundary condition for Y on boundary 9 i a no flux boundary condition. 9

11 Boundary Y u w ψ (derived from) 1 Y = Y ur u = 0 w = 0 ψ = 0(w = 0) 2 r 3 z 4 z = 0 5 z 6 r 7 z 8 r = 0 u = 0 w = 0 ψ = 0(u = 0) = 0 u = 0 w = 0 ψ = 0(w = 0) u z = 0 w z = 0 2 ψ z 2 = 0( u z = 0) = 0 u = 0 w = 0 ψ = α(w = 0) = 0 u = 0 w = 0 ψ = α(u = 0) = 0 u = 0 w = 0 ψ = α(w = 0) = 0 u = 0 w = 0 ψ = α(u = 0) 9 ρy w + Dρ z = 0 u = 0 w = c ψ = c 2 r2 (w = c) 10 r = 0 u = 0 w r = 0 ψ = 0(u = 0) Table 1: Table of boundary condition for experimental Domain 1. Phyically peaking, it enforce that no fuel vapor go in or come out of the fritted gla dik. The boundary condition in Table 1 are thoe that repreent an uncovered fuel pool open to Domain 1. Once the fuel pool i covered by a film layer, the boundary condition for Y on 1 will change to reflect Henry Law. Henry Law tate that the vapor preure of a ga i proportional to the amount of that ga diolved in a liquid. The firt validation tet that wa performed in the experimental domain wa a flux tet which involved pecifying a flux on the fuel pool urface. Thi flux boundary condition replaced the boundary condition of Y = Y ur. What thi pecified flux implied wa that there wa a contant amount of evaporation taking place on the urface. At teady tate, the amount of evaporation coming off of the urface would alo be coming out of the outlet. The code wa run out to teady tate to ee if it wa able to reproduce the evaporation at the outlet. The model wa able to do o, which meant that the extra feature in the domain were added correctly. The defining tet of the code wa to compare it to experimental data. William et al performed experiment for an uncovered n-heptane pool with a nitrogen flow rate of 78 cm3 min. The value for D A and Y ur were found repectively by uing Chapman-Enkog kinetic theory and Antoine parameter. Becaue of the relatively high flow, there wa noie in the experimental value which meant that the uncertainty range wa fairly high. The value for the experiment and the numerical model were very comparable though with the experiment meauring a total flow of 110 µg and the numerical model predicting 130 µg. With thee tet reult, it wa concluded that the model wa working correctly and could 10

12 Figure 5: Plot of the experimental Domain 1 with labeled boundarie. Thee label correpond to boundary condition in Table 1. accurately calculate concentration and total flow for the uncovered fuel pool cae. Before adding the optimization routine, Domain 2 had to be coupled to Domain 1. Phyically peaking, thi would reult in covering the fuel pool and it change ome of the previouly lited boundary condition. Table 2 i a lit of the boundary condition correponding to the labeled boundarie in Figure 6. The boundary condition of Y = Y ur on 1 now change to Y = Pv P a Y F to repreent Henry Law. In thi formulation, P v i the reulting vapor preure from a pure fuel pool open to air, P a i the atmopheric preure, and Y F repreent the fraction of fuel on the urface of the film domain. The boundary condition repreenting the fuel pool i now on boundary 11, Y = Y ur, and Y ur i the mole fraction that reult from the olubility of fuel in water. Thi i an approximation though becaue the aqueou olution ha ome amount of urfactant in it. Nonethele, thi approximation will be ued and it validity will be analyzed once the model i applied to the data. The lat boundary condition that need to be dicued i that of 13. Thi boundary condition enforce matching fluxe. The left hand ide i the flux on the top urface of the film layer in the film domain and the right hand ide repreent the flux 11

13 Boundary Y 11 Y = Y ur 12 r = 0 13 ρ F D F F z 14 r = 0 = ρ A D A A z Table 2: Table of boundary condition for experimental Domain 2. on the top urface of the film layer in the air domain. The ubcript A denote the air domain and the ubcript F denote the film domain. Once coupled, boundarie 13 and 1 will be the ame boundary. However, two boundary condition are needed ince it i the demarkation of two domain. Note that no validation i needed for Domain 2 becaue the pecie fraction olver ha already validated for a pure diffuion cae. Figure 6: Plot of the experimental Domain 2 with labeled boundarie. Thee label correpond to boundary condition in Table 2. Optimization Validation The method that wa choen to minimize the reidual between the numerical and experimental data wa the ecant method. The purpoe of the ecant method wa to find the D F value that reproduced the experimental data within ome tolerance. The data that wa provided came in everal different form. For ome experiment, the data came in the mole or ma fraction at the outlet for the uncovered and covered cae. Other ex- 12

14 periment preented the data in the form of a ratio of the two cae. Thu, to create a model that could handle both form of data, the ecant method wa deigned to operate on the ratio form. In the cae when the data wa not preented in ratio form a ratio wa eaily created by dividing the covered value by the uncovered value. The ratio format alo had the added benefit of removing ome of the experimental uncertainty from the data. ( D F,n D ) F,n 1 D F,n+1 = D F,n (R exp R n ) (R exp R n ) (R exp R n 1 ) (10) The ecant method formulation for making the next gue at D F then took the form of (10), where R exp i the ratio of the experimental data. In (10), R n and R n 1 repreent the numerical data ratio that reulted from D F,n and D F,n 1 repectively. To form R n, the uncovered cae mut be known o the model i run for the uncovered cae firt. It then tore the value in memory, cover the pool, and proceed with the covered pool imulation. Becaue the ecant method formulation i being ued, two initial guee at D F are required. For a foam layer, the initial guee were made uing Chapman Enkog kinetic theory and for a film layer, the Wilke-Chang equation for liquid-liquid diffuion i ued. Both Chapman Enkog kinetic theory and the Wilke-Chang equation for liquid-liquid diffuion are preented in [5]. To validate the implementation of the ecant method, a tet cae wa created. The tet cae concerned a 3cm high foam layer being placed on a n-heptane fuel pool. The value of D F wa initially et to D F = 0.01 cm2 o that the uncovered and covered teady tate concentration could be found. The ratio of thee two value wa et to R exp and the value of D F = 0.01 cm2 wa removed from the code. Given R exp, the model wa aked to find the D F value that would reproduce R exp R n γ where γ i ome tolerance. The model correctly reproduced D F = 0.01 cm2 for the value of R exp o the ecant method wa known to be implemented correctly. 5 Reult Application to Tet Data The firt cae run with the fully coupled, optimization capable, model wa for a n- octane pool covered by 1cm of film. The flow rate of N 2 into the container wa 630 cm3 min and at 100 the concentration wa meaured to be 0.15% of the uncovered value. To find the diffuion coefficient for n-octane in N 2, D A, Chapman-Enkog kinetic theory wa ued. Thi reulted in an etimate of D A = 0.06 cm2. A a check on the validity of uing thi kinetic theory, D A for n-octane in N 2 wa compared to experimentally determined value, which were within 3% of D A. The concentration boundary condition that wa ued on boundary 11 wa a mole fraction value of and the boundary condition ued on boundary 1 13

15 wa Henry law, 0.018Y F ( Pv P a = for n-octane). Uing thee input parameter and data, the model found that D F = cm2 wa the diffuion coefficient reponible for the experimental meaurement. Figure 7 and 8 are contour plot of the axial and radial velocitie within the container at a time of 100. Figure 9 and 10 are contour plot of the concentration (mole fraction) of fuel vapor within Domain 1 and 2 at 100. Figure 7: Contour of the axial velocity inide Domain 1 at 100 reulting from an N 2 input flow of 630 cm3 min. 14

16 Figure 8: Contour of the radial velocity inide Domain 1 at 100 reulting from an N 2 input flow of 630 cm3 min. Application to Film Data The firt cae that the model wa run on wa a et of tet data. Thi cae wa created to how that the model wa capable of modeling the experiment and finding the D F value reponible for the data. Once the model i applied to actual experimental data, 15

17 Figure 9: Contour of the fuel vapor concentration inide Domain 1 at 100 reulting from D F = cm2. it produce very intereting reult. The actual experimental data that the model wa run on wa the film layer data gathered by Leonard. The model wa not run on the foam layer data gathered by William becaue the data appear to have a converion factor error. Thu, the model wa run on the credible data that wa immediately available. One experiment that Leonard ran wa for a n-octane pool covered by cm 16

18 Figure 10: Contour of the fuel vapor concentration inide Domain 2 at 100 reulting from D F = cm2. of film. The flow rate of N 2 wa 630 cm3 min and at 1800 the concentration (mole fraction) wa meaured to be 15% of the uncovered value. The boundary condition ued on boundary 11 wa Thi value i the actual olubility of n-octane in water a oppoed to the value of that wa ued in the tet cae. The boundary condition of Henry Law, Y F, i valid for boundary 1 ince Pv P a = for n-octane and it wa applied. The firt gue of the film layer diffuion coefficient wa D F,1 = cm2. Figure 11 i a plot of the concentration (mole fraction) of fuel vapor at the outlet veru time. There are two feature of Figure 11 that need to be noted. The firt of which i that the teady tate of fuel vapor at the outlet i reached within 70 for thi value of D F,1. Changing the value of D F will only change the amount of time needed to reach teady tate and would not change the concentration of fuel vapor at teady tate. The concentration magnitude i the econd feature that need to be noted. At teady tate, the concentration of fuel vapor i on the order of 10 9, while the uncovered concentration wa found to be Taking the ratio of thee value we ee that R 1 = Comparing thee reult to Leonard data, one can ee that the time cale and ratio are both everal order of magnitude away from the numerical reult. Figure 12 i a further illutration of thi point, with Leonard data and the numerical reult for D F,1 = cm2 plotted on the ame graph. If the model wa allowed to optimize over D F, it would have gueed lower and lower value for D F. Thi would have the effect 17

19 of increaing the amount of time to reach teady tate, however, thi would require value much maller than A diffuion coefficient lower than thi value would be conitent of that of olid-olid diffuion o we can ee that value of thi order of magnitude would be unreaonable. Changing D F would alo have no effect on changing the concentration value at teady tate. With thee two thing in mind, it wa decided to perform ome analyi on the olution to ee what could be reponible for the dicrepancie. Figure 11: Plot of the fuel vapor concentration at the outlet veru time for D F 10 5 cm2. = 1 Analyi of Reult and Leonard Data In hi experiment, Leonard meaured a concentration correponding to 15% of the uncovered cae at The concentration profile at the outlet at 1800 for the uncovered cae i Y = Taking 15% of that value leave u with a meaurement of Y = for the covered cae at Noting that in Figure 9 and 10, the concentration increae a height decreae, one can expect the ame phenomena in thi cae. Thu, in Domain 1, at the film layer urface, one would expect Y > Applying Henry Law, thi time in the oppoite direction, one can ee that in Domain 2 at the film layer urface Y ur > Again, a height decreae, the concentration on fuel vapor will increae, o at the urface of the fuel pool one would expect Y ol > Y ur. Thi implie that Y ol, the mole fraction due to the olubility of n-octane in the aqueou olution, will have to be larger than , or approximately 2%. Remember that the original olubility that wa ued in thi model wa Thu, one can ee that there i a clear diagreement in the boundary condition and one that could account for the 18

20 Figure 12: Comparion of Leonard experimental data and the numerical reult for an initial gue of D F = cm2. Plot of the fuel vapor concentration at the outlet veru time. previou dicrepancie between Leonard experimental reult and the numerical reult. Figure 13: Analyi of mole fraction in Leonard experimental data. 19

21 The model ha been extenively validated and it i known that the model correctly predict the uncovered fuel pool cae. The reult captured in Figure 11 and 12, along with the above analyi, ugget that the aumption of diolution and diffuion of fuel vapor in the film layer i not valid. Thi i due to the fact that a very high olubility of n-octane in the aqueou olution i neceary in order to reproduce Leonard data. At thi point, one cannot rule out a time dependent boundary condition for Y ol nor a time dependent diffuion coefficient D F. Time dependence on Y ol and/or D F could be reponible for the large time cale oberved in Leonard data while till allowing for a diolution and diffuion tranport mechanim. Currently, emulification i being looked at a the tranport mechanim becaue of the hort coming of the diolution and diffuion mechanim highlighted by thi model. In order to try to dipel notion of diolve and diffue, experiment are being planned to meaure the olubility of fuel vapor in the aqueou olution. If the olubility i unable to reach a value of 0.02% or higher, then it will be clear that another mechanim mut be reponible for the tranport of fuel through the film layer. At the ame time, plan for rerunning the foam layer experiment are underway o that the model can be applied to the data. It will be intereting to ee if a imilar dicrepancy appear in the foam layer imulation. In the immediate future, the model will be run with Y ol 0.02 to find the effective D F value that i capable of reproducing Leonard reult. 6 Concluion At the tart of thi project, I aid that I would deliver a oftware package that modeled the experiment of Leonard and William by auming the diolution and diffuion tranport mechanim. Thi oftware package wa alo to be capable of finding an effective diffuion coefficient D F for a film or foam layer. The value of D F would be found by minimizing the reidual between the experimental and numerical reult. Unfortunately, the foam data that wa originally promied wa corrupted o the model wa only able to be run on the film layer data. Nonethele, the model that ha been created i entirely capable of imulating a foam layer. In addition to the oftware package, I aid that I would deliver my input data and it ha all been preented in thi paper. In addition to my deliverable, I have alo preented finding that ugget that emulification i the mechanim reponible for tranporting the fuel through the film layer. 20

22 7 Reference 1. Leonard, J.T., and Burnett, J.C. Suppreion of Fuel Evaporation by Aqueou Film of Fluorochemical Surfactant Solution. NRL Report William, B.A., Murray, T., Butterworth, C., Sheinon, R.S., Fleming, J., Whitehurt, C., and Farley, F. Extinguihment and Burnback Tet of Fluorinated and Fluorinefree Firefighting Foam with and without Film Formation. Suppreion, Detection, and Signaling Reearch and Application- A Technical Working Conference (SUPDET 2011). March 22-25, Orlando, Florida. 3. William, B.A, Sheinon, R.S., and Taylor, J.C. Regime of Fire Spread Acro an AFFF Covered Liquid Pool. NRL Report Ananth, R, and Farley, J.P. Suppreion Dynamic of a Co-Flow Diffuion Flame with High Expanion Aqueou Foam. Journal of Fire Science Bird, R.B., Stewart, W.E., and Lightfoot, E.N. Tranport Phenomena. John Wiley and Son Pozrikidi, C. Introduction to Theoretical and Computational Fluid Dynamic. Oxford Univerity Pre Panton, R.L. Incompreible Flow. John Wiley and Son Pre, W.H., Teukolky, S.A., Vetterling, W.T., Flannery, B.P. Numerical Recipe in Fortran. Cambridge Univerity Pre