Tethered methanol droplet combustion in carbon-dioxide enriched environment under microgravity conditions

Combustion and Flame 159 (2012) 200–209

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Combustion and Flame j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / c o m b u s t fl a m e

Tethered methanol droplet combustion in carbon-dioxide enriched environment under microgravity conditions Tanvir Farouk ⇑, Frederick L. Dryer Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ 08544, United States

a r t i c l e

i n f o

Article history: Received 28 December 2010 Received in revised form 15 April 2011 Accepted 18 June 2011 Available online 25 July 2011 Keywords: Droplet combustion Microgravity Extinction Tether Methanol Modeling

a b s t r a c t Tethered methanol droplet combustion in carbon dioxide enriched environment is simulated using a transient one-dimensional spherosymmetric droplet combustion model that includes the effects of tethering. A priori numerical predictions are compared against recent experimental data. The numerical predictions compare favorably with the experimental results and show significant effects of tethering on the experimental observations. The presence of a relatively large quartz fiber tether increases the burning rate significantly and hence decreases the extinction diameter. The simulations further show that the extinction diameter depends on both the initial droplet diameter and the ambient concentration of carbon dioxide. Increasing the droplet diameter and ambient carbon dioxide concentration both of them lead to a decrease in the burning rate and increase in the extinction diameter. The influence of ambient carbon dioxide concentration on extinction shows a sharp transition in extinction for larger size droplets (do > 1.5 mm) due to a change in the mode of extinction from diffusive to radiative control. In addition predictions from the numerical model is compared against a recently developed simplified theoretical model for predicting extinction diameter for methanol droplets, where the presence and heat transfer contribution of the tether is not taken into account implicitly. The numerical results suggest some limitation in the theoretical modeling assumptions for favorable comparisons with the experimental data. Ó 2011 The Combustion Institute. Published by Elsevier Inc. All rights reserved.

1. Introduction Droplet combustion is of interest from both fundamental and application view points. The latter is related to fire safety issues in space flights. Influences of ambient CO2 concentration on the combustion characteristics and on extinction provides insight into the fire extinguishment phenomena and effectiveness of CO2 as fire suppressant in low gravity environment. Isolated, spherically symmetric liquid fuel droplet combustion is a convenient and unique configuration to investigate relevant combustion process. The non-sooting and hygroscopic nature of methanol makes it a prospective bench mark to study different limiting cases. Although there have been a number of studies of methanol droplet combustion in the past [1–7], study on the influence of different atmospheres, especially atmosphere containing CO2 have not been conducted until recently. Cho et al. [1] conducted methanol droplet combustion experiments in He–O2 mixtures to determine extinction diameter in 2.2 s microgravity facility. The He–O2 environment enabled the measurement of the extinction diameter in this short duration microgravity environment as a result of the increased burning rate produced by the highly conductive helium ⇑ Corresponding author. E-mail address: [email protected] (T. Farouk).

mixture. Lee and Law [3] studied the combustion of free falling methanol and ethanol droplets. Their study showed that both these fuel droplets absorb water produced during the combustion process. They also measured the water content of the droplet as function of burning time and extinction diameter. Yang et al. [7] in their experimental study of methanol/dodecanol droplets showed that pure methanol droplets burning in air exhibit extinction at a non-zero droplet diameter. Okai et al. [5] studied the influence of pressure on methanol droplet combustion. Their study revealed that water absorption and dissolution in methanol droplets become less important at higher pressure. Marchese et al. [6] showed that for methanol droplets burning in air the radiative heat loss becomes noticeable for a initial droplet size greater than 1 mm. For a 3 and 5 mm droplet they report the presence of significant radiative heat loss. In a recent work Hicks et al. [8] conducted a number of microgravity drop tower experiments to study the effects of CO2 enriched environments on droplet combustion properties. It is difficult to perform free floating droplet experiments, particularly in drop tower facilities [9,10], and so these particular experiments were performed using the more generally accepted method of ‘tethering’ the droplet on a support fiber. The ‘tethering’ of droplet enables not only the ability to grow the droplet in 1-g conditions prior to the release of the drop package, but attempts to reduce

0010-2180/$ - see front matter Ó 2011 The Combustion Institute. Published by Elsevier Inc. All rights reserved. doi:10.1016/j.combustflame.2011.06.014

T. Farouk, F.L. Dryer / Combustion and Flame 159 (2012) 200–209

the effects of small disturbances produced during droplet formation, deployment and ignition that result in slow drift and degraded optical measurements of the droplet and flame diameter burning history. Two different types of materials and tethering configurations were employed in the work of Hicks et al. [8]: a single quartz fiber (dfiber = 110 lm, dbead = 360 lm), and a crossconfiguration of two silicon carbide fibers (dfiber = 10 lm, dbead = 220 lm). Each configuration affixed a bead to the fiber(s) at the tethering location to further stabilize the location of the droplet. The discussion of the experimental observations on burning rate, flame standoff, and extinction diameter note that tethering can disturb these phenomena by affecting drop shape, heat transfer between the environment and liquid, and determination of flame size and extinction diameter (as the drop approaches the bead diameter). Moreover, surface tension interaction at the fiber penetration sites also induces internal liquid phase motions that affect the gas liquid interface boundary conditions and mixing of absorbed combustion products into the drop interior. Non-spherical droplet geometries also occur as a result of tethering, and it is typical to determine ‘‘equivalent spherical diameters’’ to interpret measurements. Stating that the burning rate is what is relevant, independent of how heat transfer to the droplet occurs (gas or fiber), the theoretical analysis applied by Hicks et al. considers all modes of heat transfer to be lumped and then tests the theory [8] for the predicted relationship relating the volume ratio of extinction diameter to other parameters against the measurements:

 3 dext 1      ¼ d 1 Ko do 1 þ 4 Dw qql YY e df g

ð1Þ

f

where dext is the extinction diameter, do initial droplet diameter, K o is the average burning rate, Dw diffusion coefficient of water vapor in the gas phase between the flame and the droplet surface, ql is the liquid density, qg is the gas density, Ye is the liquid phase water mass fraction for extinction, Yf is the mass fraction of water at the d flame minus that at the surface and df is the flame standoff ratio (FSR). Predictions employing Eq. (1) will be referred to as ‘theoretical’ predictions through out this document. Though Hicks et al. [8] report ‘theoretical’ predictions of the droplet extinction diameter (dext) employing the experimental K o , the measured burning rate constants, (K o ) derived from the experiments include heat transfer effects from both fiber and gas. Thus, the proposed model cannot be used to a priori predict the dext for other tethering parameters or free-floating experiments. Moreover, the ‘theoretical’ prediction utilizes physical parameters evaluated at specified reference temperatures, and the dependence of the predictions on the choices utilized are not discussed. Very few works [9,11,12] have been reported in the literature that quantitatively investigate the consequences of tethering on combustion observations, and none have utilized fully coupled transient detailed numerical studies to quantitatively elucidate the behavior. In this work, we present ‘numerical’ modeling results for the Hicks et al. tethered methanol droplet experiments using a recently developed spherosymmetric droplet model [13] that includes implicitly the presence of a tether fiber in the computations. Comparison is drawn between the experimental and the two model predictions; ‘numerical’ and ‘theoretical’. A priori predictions from the ‘numerical’ model compare favorably with the experimentally measured droplet and flame diameter evolution, average burning rate (K o ) and dext. The ‘numerical’ model also permits an assessment of parameter effects, and further insights into the predictions yielded by the ‘theoretical’ model. In addition, modeling results on the influence of ambient CO2 for different initial droplet sizes are further elucidated by accounting for the presence of the fiber supports.

201

2. Mathematical model The model that we employed here is described in detail in [13,14], and we will only briefly summarize its features here. The model utilizes detailed gas phase kinetics, spectrally resolved radiative heat transfer, and multi-component transport in solving the species and energy conservation equations for both the liquid and the gas phase. Coupling of the liquid and gas interface satisfy thermodynamic constraints and conservation of material fluxes. Gas phase radiant interactions are incorporated using a statistical narrow band (SNB) radiation submodel following the exact treatment of Ju and coworkers [15]. Additional submodels are used to evaluate the density, diffusion velocity in both phases and chemical reactions in the gas phase. The mathematical model provides a spherosymmetric representation of the droplet combustion phenomena. The model consists of species and energy conservation equations in both phases having the following expression:

Z

Z rþ r 2 r_ jrþ @ðyi qÞ 2 y qr 2 dr þ yi ðqu  qr_ Þr 2 jrrþ r dr þ 1 3 rrþ @t r jr  r  i r 3 Z rþ ¼ yi qV i r 2 jrrþ þ x_ i r2 dr rþ

ð2Þ

r

Z

Z rþ r 2 r_ jrþ @ðhqÞ 2 r dr þ 1 3 rrþ hqr 2 dr þ hðqu  qr_ Þr2 jrrþ @t r j r r  r 3 r þ ¼ qr 2 r þ Sloss=gain due to fiber rþ

X

yi ¼ 1

ð3Þ ð4Þ

8i

where yi = mass fraction, q = mass density, r = radius, u = bulk fluid velocity, r_ = velocity of the control volume boundary, Vi = diffusion _ i = rate of species production due to chemical reaction, velocity, x h = enthalpy per unit mass and q = heat flux. The heat flux q represents here a lumped term, with contributions from chemical enthalpy of diffusing species, thermal conductive heat transport, radiative heat transport and the Dufour effect. It should be noted that the second term on the left hand side of both Eqs. (2) and (3) is the volume dilation term and represents the growing or shrinking of the infinitesimal volume as it follows the motion. Sloss/gain due to fiber is the loss and gain due to the presence of the tether fiber. Since the fiber is only present in a limited region the loss/gain term associated with the fiber is expressed in terms of ratio of the fiber and domain volume and has the following expression:

Sloss=gain due to fiber ¼ Afiber ¼ pr 2fiber

Z

 4hconv ;j nf Afiber lfiber ðT  T f Þr2 dr ; dfiber Vj r 4 3 and V j ¼ prj 3 rþ

ð5Þ

where the subscript j denotes gas or liquid phase, hconv,j is the convection coefficient and is hconv,liq/hconv,gas depending on the location of the fiber. For r  6 r 6 0, the fiber is immersed in the droplet and hconv,j = hconv,liq and for 0 6 r 6 rþ , the fiber is in the gas phase and hconv,j = hconv,gas. The term nf represents the number of fibers (i.e. nf = 4 for a cross string arrangement and nf = 2 for a single fiber arrangement), dfiber is the fiber diameter, T is the temperature in the liquid/gas domain, Tf is the fiber temperature, Afiber is the fiber cross sectional area, lf is the length of the fiber, Vj is the volume of the liquid or gas phase domain and rj is the liquid/gas phase radius. This loss/gain term provides a two way coupling between the gas/ liquid phase energy conservation and the energy conservation in the fiber. It should be noted that Sloss/gain due to fiber acts as a source or a sink term depending on the difference between T and Tf (sink term for T > Tf and source term for T < Tf).

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Equations at the liquid–gas interface arise either through satisfaction of thermodynamic constraints or conservation of material fluxes:

yi ðqu  qr_ Þjþ ¼ yi qV i jþ X

ð6Þ

hv ap;i ½yi ðqu  qr_ Þ þ yi qV i þ ¼ qjþ  qfiber

ð7Þ

i2liquid

X i2gas

yi;þ ¼ 1;

X

yi; ¼ 1;

and T þ ¼ T 

ð8Þ

i2liquid

xi;þ p ¼ xi; ci ðT; xj; Þpv ap;i ðTÞ

ð9Þ

where hvap,i = enthalpy of vaporization, p = system pressure, T = temperature, xi = mole fraction, ci = activity coefficient and pvap,i = vapor pressure of the ith component in its pure state. The symbols ‘+’ and ‘’ denote location in the gas and liquid phase, respectively. The heat flux q at the interface has contribution from chemical enthalpy of diffusing species, thermal conductive heat transport, radiative heat transport and Dufour effect. The qfiber represents the heat flux through the fiber at the interface. The heat transfer effect through the tether fiber is included using a transient, one dimensional model approximation. Due to the small size of the fibers employed in the experiments [8] (dfiber = 10 lm and dfiber = 110 lm) and the associated small Biot numbers, the temperature distribution in each fiber can be assumed to be one dimensional. Assuming a black body radiation, the energy conservation equation in the fiber is given by:

  @ðqf cp;f T f Þ @T f @ 4hconv ;j þ kf ðT  T f Þ ¼ @x @t @x dfiber  4ref  4  T f  T 41 dfiber

ð10Þ

qf is the fiber density, cp,f is the specific heat of the fiber, kf is the fiber thermal conductivity, Tf is the fiber temperature, dfiber is the fiber diameter, r is the Stefan–Boltzmann constant, ef is the fiber emissivity, T1 is the far field temperature, T is the temperature adjacent to the fiber (Fig. 1) in the liquid/gas domain and hconv is the convection coefficient. The second term in the right hand side represents the convective heat flux from the gas and liquid surrounding to the fiber and vice versa depending on Tf being lower or higher than T. The third term represents the radiative heat flux from the fiber. It should be noted that the radiant emission from the fiber is only included in the domain where the fiber is exposed to the gas phase. The  convection coefficient hconv,j is obtained with hconv ;j ¼ Nu1 kj =dfiber j¼liq=gas , assuming the fiber to be horizontal cylinder. With Tf solved, the heat flux through the fiber qfiber can be determined. A schematic diagram illustrating the fiber domain, fiber grids, grid stencil together with the convective and radiative flux contributions is presented in Fig. 1. In the model the volume boundaries are defined to coincide with the liquid–gas interface and the far field (two hundred times the initial droplet diameter) boundary is well defined using Dirichlet conditions and remains fixed in the simulations. The Dirichlet conditions imposed on the far-field are of fixed ambient composition (fixed O2, CO2 and N2 concentration) and temperature. The inner most liquid node is centered at the origin, providing the required no-flux condition. Similarly a Dirichlet and zero flux condition are imposed on the far field and the fiber located at the origin. For the Dirichlet boundaries the temperature was prescribed to have an ambient temperature (T1) of 298 K. Several Nusselt number correlations were applied in the simulations that encompass stagnant conditions [16] as well as low Reynolds number flow [17–19] conditions for cross and parallel flow over a cylinder. The Nusselt number solution of Mahmood and Merkin [19] was employed in the simulations which was

Fig. 1. Schematic of the fiber with the numerical grids with an inset of grid stencil for three node centered grid points. The subscript ‘m’ and ‘p’ denotes some arbitrary grid location. qw and qe denotes flux at the faces (west and east for the m and m + p location).

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For the larger quartz fiber the Nusselt number at the droplet surface for the different correlations were:

obtained for convection over a horizontal cylinder with the flow field being parallel to the cylinder axis. The formulation was based on boundary layer equations. Mahmood and Merkin’s obtained   1=2 Nu Re their Nusselt number solution as Re1 r vs: d r Re1 cylinder . In the cylinder

Nu1;Andrews ¼ 0:750; Nu1;Collis ¼ 0:588;

simulations, Rer which is Reynolds number along the axial direction of the tether was determined at each location (grid point) depending on ur, mr and the radial distance r (the axial direction of the tether coincides with the radial direction of the droplet). Similarly Recylinder which is the Reynolds number for the fiber having the fiber diameter as the characteristics length was determined at each location as well. For Recylinder only ur and mr varied. The Rer, Recylinder and Nua number were determined at each grid point location and at every time instance with the assumption that each small segment of the fiber acts as an isothermal cylinder at that specific time. The radial velocity ur was determined from the Stefan flux which decreases to smaller values with increasing distance. Nusselt number correlation of Churchill and Chu [16]

The large Nusselt number is confined to a very small region close to the droplet liquid/vapor interface and decreases rapidly as the Reynolds number rapidly decreases to smaller values with increasing radial distance into the gas phase. The bulk of the fiber experiences a near-stagnant condition. The small Reynolds number and the exposure of the bulk of the fiber to stagnant conditions results in a minimal response to the choice of Nusselt number correlation in terms of predicted behavior. The results reported here correspond to utilizing the stagnant Nusselt number correlation. Predictions employing the transient, spherosymmetric mathematical model will be referred to as ‘numerical’ predictions through out this document.

(Nu1 = 0.36) Andrews et al. [17] (Nu1 ¼ ð0:34 þ 0:65Re0:45 Þ)   cylinder  T m 0:17 and Collis and Williams [18] Nu1 ¼ 0:24 þ 0:56Re0:45 ; cylinder T  TþT f Tm ¼ 2 were also employed which are valid for cross flow

3. Results and discussions Simulations were carried out for each of the individual experiments reported in [8]. The experimentally measured and predicted K o and dext together with the individual experimental conditions are presented in Table 1. The same experiment designation numbers as reported in [8] were used for the purpose of consistent referencing. The simulations were conducted by employing the methanol oxidation mechanism of Li et al. [20] consisting of 21 species undergoing 93 reactions. The data correlations of Daubert and Danner [21] were used to calculate the liquid properties. The simulations employed temperature-dependent variable thermal conductivity [22,23] and constant density and specific heat of qf,quartz = 2200 kg/m3, cp,quartz = 760 J/kg K and qf,SiC = 2740 kg/m3, cp,SiC = 670 J/kg K for the quartz and SiC fibers, respectively. Table 1 shows that for all the cases the numerically predicted K o and dext compare favorably with the experiments. The deviations between the predicted and measured K o were found to be 5%. A higher discrepancy was observed between the measured and numerically predicted dext, with the prediction being smaller than the experiments for all the cases. The maximum discrepancy observed for dext was found to be a factor of 1.5. Since large beads were used in the experiments, especially for those with quartz

conditions. The Nusselt number for each grid location was also determined when the cross flow correlations was used based on the same assumption, except for the Churchill and Chu correlation. From Nua the convective coefficient at each grid point was determined by employing the thermal conductivity value at that specific grid location which could be in gas or liquid phase. Tliq, Tgas, hconv,gas and hconv,liq are computed at each grid location (Fig. 1) which influences the tether temperature at that specific location. In this fashion the tether temperature at the center line responds in a non-linear fashion to the non-uniform temperature field along its surface [13]. However depending on the choice of the correlation the burning history, flame standoff ratio and extinction diameter prediction varied by 3% between them. The variation in the overall results due to the choice of the Nusselt number was not significant due to the small Reynolds number of the flow. For the SiC fiber the Nusselt number at the droplet surface for the different correlations were:

Nu1;Andrews ¼ 0:465; Nu1;Collis ¼ 0:357;

Nu1;Mahmood

¼ 0:710 and Nu1;Churchill ¼ 0:36

cylinder

Nu1;Mahmood

¼ 0:195 and Nu1;Churchill ¼ 0:36 Table 1 Experimental conditions, measured and predicted burning rate and extinction diameter. Experiment Nos.

Fiber

CO2 (%)

do (mm)

K oexpt (mm2/s)

dextexpt (mm)

dbead (mm)

dextcorr (mm)

K ocalc (mm2/s)

dextcalc (mm)

H73 H74 H75 H77 H78 H79 H80 H83 H87 H95 H113 H115 H120 H121 H124 H125

Q Q Q Q Q Q Q SiC SiC SiC SiC SiC SiC SiC SiC SiC

27 70 70 70 51 51 55 51 51 0 70 25 25 35 15 75

1.72 1.66 1.50 1.63 1.38 1.48 1.54 1.70 1.39 1.44 1.51 1.56 1.25 1.44 1.34 1.40

0.61 0.54 0.56 0.52 0.58 0.60 0.55 0.50 0.51 0.54 0.46 0.53 0.53 0.50 0.57 0.51

– – 0.65 0.74 0.60 0.62 0.68 0.91 0.81 0.70 0.72 0.78 0.47 0.60 0.64 0.64

0.360

– – 0.61 0.71 0.55 0.58 0.64 0.90 0.80 0.69 0.71 0.77 0.45 0.59 0.63 0.63

0.60 0.51 0.53 0.50 0.61 0.60 0.55 0.48 0.49 0.56 0.44 0.51 0.55 0.51 0.56 0.48

0.49 0.68 0.51 0.63 0.41 0.44 0.48 0.67 0.55 0.44 0.65 0.53 0.41 0.51 0.43 0.60

0.220

Experiment nos. = test nos. of Hicks et al. [8]. Experiments were conducted for a fixed oxygen concentration of 21%, a variable carbon dioxide concentration as indicated and balance of nitrogen. Q = Quartz fiber, dfiber = 110 lm, single fiber arrangement. SiC = Silicon-carbide fiber, dfiber = 10 lm, cross string arrangement. qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 dextcorr = Corrected extinction diameter, dextcorr ¼ ½ðdextexpt Þ3  ðdbead Þ3 .

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fibers, a ‘corrected’ extinction diameter (dextcorr) was determined by subtracting the bead volume from the experimental extinction droplet volume. The dextcorr value was smaller by 7% for the quartz fiber cases (H73–H80) due to the fact that a larger bead was employed and smaller dext were observed. The deviation for dextcorr for the SiC tether cases were minimal (H83–H125) which is the consequence of the small bead and large dext observed. Subtracting the bead volume from the measured extinction volume provides a conservative estimate of the error bounds of the extinction diameter measurements. The effects of the bead geometrical location at extinction and uncertainties in the reported extinction diameters are not discussed in [8]. It was however reported [8] that the presence of the bead introduced significant perturbations in the experiments during the extinction stages. Even for the smaller sized bead employed in the SiC tether arrangement, it was found that the bead was slightly porous, which could easily have entrapped gas and so perturb the dext measurements. Moreover, we note that the presence of the fiber and the beads can affect induced internal liquid phase motions as the extinction is approached, a phenomena not taken into account in the ‘numerical’ model. The experimental results are further complicated by the fact that at extinction, the droplet was not centered on the beads, but frequently moved to a location where the bead interfaced with the fiber and was only partly submerged within the droplet, especially for the quartz tether configuration. In summary, the observed discrepancy between the measured and numerically predicted dext is most likely due to the presence of the stabilizing fiber beads. The measured and predicted evolution of the droplet diameter, Ko and flame radius for experiments H113 and H75 is presented in Fig. 1. Simulation result for a tether fiber-less scenario is also presented. Experiment H113 and H75 were the preferred choice as they had almost identical conditions except for the tether fiber and arrangement (Table 1). The experimental droplet diameter evolution history of Hicks et al. [8] were corrected for the shape distortion following the methodology of Struk et al. [11] and were reported as an equivalent spherosymmetric diameter. It is clearly evident that the larger quartz tether fiber significantly increases the droplet diameter regression rate in comparison to free droplet observations. The predicted burning rate constant with quartz fiber present is 23% larger (resulting in a smaller dext and shorter burn time) in comparison to fiber-less predictions. In contrast, the SiC tether due to its small size (a factor of 11 smaller than the quartz fiber) was found to have minimal effect on the droplet diameter regression rate, closely following the same trend as that of the predictions without a fiber present. The increase in Ko due to the SiC fiber was 2.2%. The Ko evolution (Fig. 2a) shows that a quasi-steady burn is achieved even with the presence of a SiC fiber. However the quartz tether induces an increased non-linearity in Ko towards the end of the burn as the regressing droplet size becomes comparable to the fiber size. This is apparent in the increase of Ko during the final stage of the burn which is not observed for the SiC fiber case. The experimental flame diameter was visually determined from video images and therefore estimations cannot be directly related to any specific parameter observed in the ‘numerical’ predictions. The predicted flame diameter evolution was estimated using the location of maximum temperature (Tmax) as a marker for the flame position. Several other parameters were investigated in prior work [13], with best agreement achieved using Tmax. The flame structure evolution (Fig. 2b) shows that the flame diameter (dflame) rapidly increases in size, plateaus for a short time and then responds to the regressing droplet size. The predictions from the ‘numerical’ model compare favorably to the measurements, and the discrepancies are well within the experimental reported uncertainties (8%). It can be clearly seen that the presence of a quartz tether fiber significantly increases the flame diameter, consistent with the higher Ko that was experimentally observed.

Fig. 2. Measured and numerically predicted evolution of (a) droplet diameter and burning rate and (b) flame diameter for a methanol droplet at atmospheric pressure under microgravity conditions. The simulation conditions are identical to the experimental conditions of H113 and H75 (Table 1). The simulations for the fiberless case had identical conditions as that of the experiment H113 without the presence of the tether fiber. The experimental data are that of Hicks et al. [8].

Tethered microgravity droplet combustion experiments for ethanol and decane [24,25] have shown that no radiant emission from relatively large quartz fibers is observed. Radiant emission from SiC tether fibers is important and is observed in many droplet combustion experiments [26] and in some cases emission for SiC tether fibers are also used to determine the flame diameter location [24]. However there is lack of availability of accurate emissivity data for fiber materials which is required for the mathematical modeling. It is therefore worthwhile to study the influence of the emissivity value on the burning history, flame diameter evolution, average burning rate and extinction diameter predictions. Figure 3 presents the influence of radiant emission from the fiber on the droplet diameter regression and flame diameter evolution for conditions identical to the experiments H113 and H75. It can be clearly seen that radiant emission from the relatively large quartz tether fiber significantly influences the droplet diameter regression as well as the flame diameter evolution. Due to the radiative heat loss from the tether fiber the heat flux through the fiber to the droplet reduces, thereby resulting in a decrease in the droplet diameter regression rate as well as flame diameter size. For the SiC tether fiber any radiant loss from the fiber had very minimal effect. The droplet diameter regression and flame diameter evolution are indiscernible from the case which did not include radiative heat loss from the fiber. The small sized SiC fibers employed in the experiments have very small heat feedback to the droplet

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Fig. 3. Effect of fiber emissivity on droplet diameter regression and flame diameter evolution for quartz (dfiber = 110 lm, single fiber) and SiC (dfiber = 10 lm, cross string) tether arrangement (do = 1.5 mm, 21% O2, 70% CO2 and 9% N2, atmospheric pressure). Comparison is shown for two extreme cases e = 0 (no radiative loss from fiber) and e = 1 (black body radiative loss from the fiber).

(SiC tether burning rate is very close to tether less burning rate), the radiative loss from the fiber further decreases the heat flux through the fiber to the droplet. In addition, the smaller SiC tether fiber has less radiative heat loss in comparison to the relatively larger sized quartz tether. Figure 4 shows the effect of ef on K o (Fig. 4a) and dext (Fig. 4b) for the two different tether arrangements. The plots show that the values of ef have very minimal effect on K o (Fig. 4a) and dext for the cross string SiC tether arrangement. For the SiC tether K o and dext were found to have a maximum variation of 0.2% and 0.3% over the range of ef values studied. The small sized SiC fiber has a very minimal heat feedback to the droplet and nominal radiative heat loss which results in the negligible variation of K o and dext. It can be seen that the quartz tether fiber has larger variation in K o and dext over the range of ef values, due to its relatively larger size. Increasing ef increases radiative loss from the quartz tether fiber which reduces the heat input from the fiber to the droplet. Over the range of the ef values K o for the quartz fiber was found to decrease almost linearly with increasing emissivity. An inverse trend was observed for dext. In between ef = 0 and ef = 1, K o and dext for the quartz fiber was found to have a maximum variation of 6% and 8% respectively.

205

The effect of CO2 concentration in the ambient gas on droplet diameter regression, Ko, FSR and peak gas temperature for a fixed initial droplet diameter of 2.0 mm is summarized in Fig. 5. The droplet diameter regression rate decreases with an increase in the ambient CO2 concentration, which results in a lower Ko (Fig. 5b). Increasing the concentration of CO2 reduces the average thermal conductivity and also increases specific heat and hence the heat capacity of the surrounding gas mixture, compared to that with N2 diluent alone. The collective result is diminished heat feedback from the flame to the droplet, reducing the droplet burning rate constant, Ko. With increasing CO2 content in the ambient gases, a quasi-steady burn was still attained as can be seen in the Ko evolution plot. For a 75% CO2 concentration the droplet burns for only a very short time and then extinguishes as a result of excessive radiative losses. The predicted FSR time histories remain essentially constant during most of the droplet life time, which further reinforces that quasi-steady burn is achieved. The predicted FSR (5.0) is only weakly dependent on CO2 concentration, which is consistent with experimental observations [8,27]. In microgravity liquid droplet combustion, the flame locates itself in the region where the fuel and oxidizer are at the stoichiometric condition. Increasing the CO2 concentration decreases Ko but at the same time it decreases the diffusion of oxygen as well, which results in this minimal variation of FSR with increasing CO2. The temporal evolution of the peak gas temperature as a function of CO2 concentration is presented in Fig. 5d. It can be seen clearly that as CO2 concentration increases, in addition to the decrease in the maximum value of the peak gas temperature,   the maximum rate of increase of the peak gas temperature dTdtmax max also decreases. In comparison to N2, the higher molecular weight CO2 has a lower thermal conductivity and higher heat capacity and is the reason for the changes observed in the early stages of the temperature evolution. During the quasi-steady burn period the peak gas temperature for the higher CO2 concentration decreases at a faster rate, due to the fact that CO2 is radiatively participating (while the displaced N2 was not). Figure 6 represents K o and dext as a function of ambient CO2 concentration for a range of initial droplet diameters. The measurements and ‘numerical’ predictions for quartz (Fig. 6a and c) and SiC (Fig. 6b and d) tether arrangements are presented separately. It can be seen that an increase in the CO2 concentration reduces K o . This is due to the fact that the increasing amounts of CO2 reduce the average thermal conductivity of the mixture compared to N2 and as a consequence the energy fed back to the droplet from the flame is diminished which results in a lower K o . K o decreases

Fig. 4. Effect of tether fiber emissivity on (a) average burning rate and (b) extinction diameter for the quartz and SiC tether arrangements.

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Fig. 5. Numerically predicted evolution of (a) droplet diameter (b) burning rate (c) flame standoff ratio and (d) peak gas temperature for a methanol droplet in varying ambient carbon-dioxide concentration (do = 2.0 mm, 21% O2, varying CO2/balance N2, atmospheric pressure, SiC tether fiber, dfiber = 10 lm, cross string arrangement).

almost linearly for small initial droplet diameters (do 6 1:50 mm). For larger droplets a non-linearity is observed for increased CO2 concentration due to increased radiative losses. The prominence of gas phase radiative losses is clearly depicted in Fig. 6c and d, where sharp changes in dext can be seen for larger droplets at increased CO2 concentration. The K o predictions (Fig. 6a and b) show that due to its larger size, the quartz fiber produces higher K o for the same initial droplet size, as a consequence of higher heat conducted through the fiber to the liquid. The predicted trends of K o agrees favorably to the experiments. The scatter in the experimental data of K o is due to the different initial droplet diameters employed in the experiments. The dext predictions (Fig. 6c and d) show an increase with increasing CO2 concentration. For small initial droplet diameters (do 6 1:5 mm) dext varies almost linearly with CO2; this denotes the diffusive extinction region. For larger droplets, a transition from a linear increase of dext is observed, similar to that noted in the work of Kazakov et al. [28]. The transition occurs as radiative losses increase with increasing droplet diameter. This sharp transition in dext for initial droplet diameters of 1.75 mm, 2.0 mm, 2.5 mm and 3.0 mm occurred for CO2 concentration of 75%, 50%, 30% and 25% respectively. The results clearly show that the large quartz fiber tether produces smaller extinction diameter than the SiC fiber because of the enhanced heat conduction to the droplet. In comparison to the quartz fiber the predicted dext for the SiC fiber were found to be larger by a factor of 1.2. It should be noted that for a do of 1 mm the quartz fiber tether resulted in a complete burn of the droplet and no extinction was observed. The

measured dext as a function of CO2 agrees favorably to the predicted trends of the ‘numerical’ model and most of them lie in the diffusive extinction region. Comparison between the experimental and predicted dext for both the ‘theoretical’ and ‘numerical’ model is presented in Fig. 7  3   and DKwo suggested by Hicks using the scaling of variables ddexto  3 and coworkers. Figure 7 shows a non-dimensional plot of ddexto    1 vs. DKwo . The ‘theoretical’ prediction using Eq. (1) shows a linear  3   relationship between ddexto  1 and DKwo indicating a linear decrease in the extinction volume with increasing average burning rate. Three different gas phase densities were chosen to evaluate the quality of ‘theoretical’ predictions using Eq. (1), qg evaluated at 337 K (boiling temperature of methanol), 1000 K and 1500 K. The 337 K reference temperature for qg was chosen to follow the exact methodology of Hicks et al. [8]. The reference temperature of 1000 K and 1500 K for qg evaluation was chosen as such as Hicks and coworkers scaled the experimental burning rate by Dw evaluated at 1000 K and the average flame temperature for methanol droplet in CO2 environment is 1500 K (Fig. 5d). For these three cases Ye = 0.3, Yf = 0.1, df/d = 5.5 and ql is evaluated at 337 K as proposed by Hicks and coworkers. It can be seen that the ‘theoretical’ predictions using qg evaluated at 337 K passes through the center of the cluster of data points. It should be noted, however, that Hicks et al. [8] suggests using Dw evaluated at an average temperature of

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Fig. 6. Comparison between measurements and ‘numerical’ prediction of (a) average burning rate for a quartz tether fiber (b) average burning rate for a SiC tether fiber (c) extinction diameter for a quartz tether fiber and (d) extinction diameter for a SiC tether fiber for a varying ambient carbon-dioxide concentration for different initial methanol droplet sizes. The experimental data are that of Hicks et al. [8].

1000 K in their ‘theoretical’ model. The rational choice of gas density is therefore not that evaluated at 337 K, but at 1000 K. Eq. (1) is highly sensitive to the choice of the gas density and changes significantly for a gas density evaluated at 1000 K or 1500 K as can be seen in Fig. 7a. Employing qg evaluated at 1000 K and 1500 K the ‘theoretical’ predictions consistently over-predicts the experimental (do/dext)3 data (i.e. under-predicting the experimental dext). Even though Hicks and coworkers suggested using the experimentally obtained K o to predict dext for each of the experiments using the ‘theoretical’ model, the ‘theoretical’ extinction diameters predictions are not reported in [8]. In addition to the predictions from the ‘theoretical’ model the ‘numerical’ predictions for initial droplet diameter of 1.5 mm and 1.75 mm for both quartz and tether fiber arrangement is also presented in Fig. 7a. Majority of the initial droplet sizes of the experiments were in this range, hence simulation results for these bounding initial diameters are presented. The simulations were conducted for increasing CO2 concentration in the ambient. Since Eq. (1) relates the extinction volume to the non-dimensionalized burning rate, the ‘numerically’ predicted extinction diameter was corrected to take into account the presence of tether bead. The presence of the bead inside the droplet increases the extinction volume. The respective bead volume   3 V bead ¼ p6 dbead was therefore added to the ‘numerical’ predictions to obtain an equivalent extinction volume assuming the bead to be completely covered by the droplet during extinction. The predic-

tions from the ‘numerical’ model show a sharp change in the extinction diameter over a small burning rate range and the experimental observation is found to be within the banded region of the prediction. To get further insight, the experimental and predicted extinction diameters from the ‘numerical’ model are presented as a function of the average burning rate in Fig. 7b, which is one of the main observable features and governing parameter of the droplet combustion phenomena. To obtain a clear trend in the extinction characteristics, numerical simulation results having do 1.5 mm and 1.75 mm are presented. The average burning rate variation was obtained by varying the ambient CO2 concentration from 0% to 75%, having O2 fixed at 21% and with a balance of N2. It can be clearly seen that the quartz fiber results in a higher K o and smaller dext. The dext is found to increase with decreasing K o (a consequence of increase CO2 in the ambience). For higher K o a linear increase in the extinction diameter is observed, denoting the diffusive extinction region. In the diffusive extinction region where the extinction diameters are small the presence of a bead would induce significant perturbations. The K o decreases with increasing CO2, as CO2 is radiatively participating and results in larger extinction diameter. This transition is not evident for do of 1.5 mm but clearly visible for do of 1.75 mm. For do of 1.75 mm a distinct non-linear response of the dext to the varying K o is observed due to the gas phase radiative losses. It was shown by Marchese and Dryer [6] that for extinction occurring in the radiative region the

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 3   Fig. 7. (a) Comparison between the experimental extinction data, ‘theoretical’ and ‘numerical’ predictions presented using the scaling of variables ddexto and DKwo suggested by Hicks et al. [8]. The gas phase density in Eq. (1) was evaluated as (a) air at 337 K (b) air at 1000 K and (c) air at 1500 K. Simulations for 1.5 and 1.75 mm initial droplet diameter were conducted for varying CO2 concentration (0–75%), 21% O2 and balance of N2. The experimental data are that of Hicks et al. [8]. (b) Comparison between the experimental and ‘numerically’ predicted extinction data presented as a function of the average burning rate.

water diffusivity Dw is not the governing parameter. Even though the ‘numerical’ predictions under predicts dext it captures the qualitative trend of the observed experimental extinction characteristics. 4. Conclusions Fiber tethered methanol droplet combustion in CO2 under microgravity conditions have been successfully simulated using a recently developed transient spherosymmetric fully coupled droplet and tether fiber ‘numerical’ model. A priori predictions from the ‘numerical’ model have been compared against the experimental data set of Hicks et al. [8]. The ‘numerical’ predictions were found to agree very well with the experimental measurements. The presence of a tether, especially the thick quartz fibers, was found to significantly increase the burning rate and the flame diameter and to decrease the extinction diameter compared to untethered observations. The large quartz fiber was found to induce a slight non-linearity in Ko towards the end of the burn when the droplet size becomes comparable to the fiber size. Smaller diameter SiC fiber tethering was found to have minimal effects yielding predictions nearly the same as for untethered conditions. Increasing the ambient CO2 concentration was found to decrease the burning rate and increase the extinction diameter. The influence of ambient CO2

concentration on extinction shows a sharp transition in dext for larger size droplets (do > 1.5 mm) due to a change in the mode of extinction from diffusive to radiative control. The transition for initial droplet diameters of 1.75 mm, 2.0 mm, 2.5 mm and 3.0 mm occurred for CO2 concentration of 75%, 50%, 30% and 25% respectively. Predictions of extinction diameter from the numerical simulations and theory of Hicks and coworkers were compared against the experimental data. ‘Numerical’ simulations were found to reproduce the burning rate very accurately, but to under-predict extinction diameter. The predictions using Hicks and coworkers theory was found to depend strongly on the reference temperature for evaluating the gas density. Gas density evaluated at 337 K showed good agreement with the experiments, however gas density evaluated at 1000 K, the average temperature between the flame and the droplet surface (used for evaluating other properties), increased the deviation from the experimental data. Acknowledgments The authors are pleased to acknowledge the financial support of National Aeronautics and Space Administration (Grant Nos. NNCO4AA66A, NNX09AW 19A). Special thanks to Mr. Michael Hicks and Dr. Vedha Nayagam for providing the experimental data for H75 and H113.

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