Supporting material
Calculation of theoretical burning constants
Following Law and Williams 32,71 the burning constant is given as
𝑘=
8𝜆𝑔
ln(1 + 𝐵)
𝜌𝑙 𝑐𝑝𝑔
𝐹
𝑌∞ (𝑂) Δr 𝐻 − 𝑐𝑝𝑔 (𝑇𝑏𝑝 − 𝑇∞ )
𝐵=(
)
Δ𝑣 𝐻
where the thermal conductivity 𝜆𝑔 and isobaric heat capacity 𝑐𝑝𝑔 are evaluated at the
mean temperature 𝑇𝑚 = (𝑇𝑏𝑝 + 𝑇𝑓 )/2 between the flame temperature 𝑇𝑓 and the boiling
point 𝑇𝑏𝑝 of the liquid fuel. The specific heat capacity 𝑐𝑝𝑔 of the vapor is taken as the
solvent vapor and 𝜆𝑔 = 0.4𝜆𝑓𝑢𝑒𝑙 + 0.6𝜆𝑜𝑥𝑦𝑔𝑒𝑛 as the average of the solvent and oxygen.
Since the flame temperature is difficult to measure, 𝑇𝑓 ≈ 3100𝐾 is taken as a reasonable
average value for all solvents, as calculated with the Web-Code from McBride and
Gordon 85 from NASA Lewis Research Center. The heat of vaporization Δ𝑣 𝐻 and the
liquid density 𝜌𝑙 are taken at the boiling point 𝑇𝑏𝑝 of the liquid solvents. The heat of
combustion Δ𝑟 𝐻 is calculated at the ambient oxygen temperature 𝑇∞ = 298.15𝐾 from the
difference in the heats of formation, assuming the general combustion reaction:
𝑋 + 𝑂2 → 𝐶𝑂2 + 𝐻2 𝑂
where X is the fuel. The correlations for 𝜆𝑔 , 𝑐𝑝𝑔 , and 𝜌𝑙 are taken from VDI Heat Atlas 86,
since the correlations in Reid et al. 87 give unreliable values for 𝑐𝑝𝑔 of ethanol and 𝜆𝑔 of
xylene.
The following correlation is used for the liquid density [kg/m³]
𝜌𝑙 =
𝐴
𝑇 𝐷
1+(1− 𝑚 )
𝐶
𝐵
where 𝑇𝑚 [°K] is the mean temperature. The relevant constants are as follows.
Liquid
m-xylene
ethanol
heptane
dodecane
A
3.8870169
1.6432653
5.2745973
1.7011294
B
0.06065
0.04088
0.07741
0.04326
1
C
636.78
529.625
557.342
682.388
D
0.12093
0.09809
0.13673
0.11512
The specific heat capacities [J/(kg K)] are calculated with
0
𝑐𝑝𝑔
= 𝐴 + 𝐵𝑇𝑚 + 𝐶𝑇𝑚2 + 𝐷𝑇𝑚3 + 𝐸𝑇𝑚−2
where the constants are as follows.
Liquid
m-xylene
ethanol
heptane
dodecane
A
-155.2
116.9
-104.9
-4.5
B
5.36
5.125
6.982
6.539
C
-0.002773
-0.002685
-0.003907
-0.003487
D x 10³
0.000511
0.000545
0.000844
0.000713
E x 10³
0.000317
0.00088
-0.000154
-0.000226
D x 109
-0.06054
0.02568
E x 1012
0.05839
-0.00504
The heat conductivities [W/(mK)] are calculated with
𝜆𝑔 = 𝐴 + 𝐵𝑇𝑚 + 𝐶𝑇𝑚2 + 𝐷𝑇𝑚3 + 𝐸𝑇𝑚4
where the constants are as follows.
Liquid
m-xylene
ethanol
heptane
dodecane
oxygen
A x 10³
-30.34
-2.69
-5.44
-7.22
-1.29
C x 106
0.00947
0.1096
0.08802
0.07379
-0.05263
B x 10³
0.123
0.027
0.033
0.027
0.107
The values of Δ𝑣 𝐻 and Δ𝑟 𝐻 are taken from CRC 83. 𝑌∞ and (F/O) are the oxygen mass
fraction in the ambience and fuel-to-oxidizer mass ratio, respectively.
Preparation of SnO2 nanoparticles
The 𝑆𝑛𝑂2 nanoparticles are prepared by FSP with 0.5 mol/L Sn(II)2-ethylhexanoate in
xylene at common FSP parameters (e.g. 5 L/min dispersion gas at 2.5 bar absolute, 5
mL/min liquid feed rate, and methane/oxygen at 1.5/3.2 L/min). The Brunauer-EmmettTeller and X-Ray diffraction diameters are 9.1 nm and 9.3 nm, respectively. To avoid
additional droplet disruption, the 𝑆𝑛𝑂2 nanoparticles are only suspended with the aid of
an ultra-sonic bath for 2 min. Although sedimentation of the larger and agglomerated
nanoparticles can occur, the smaller nanoparticles stay suspended over a sufficiently
long time to conduct the experiments.
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Example of flame contraction of a 0.5 mol/L Sn(II)2-ethylhexanoate in xylene
solution
Figure A: Evolution of (a) averaged flame diameter and normalized velocity and (b)
burning constant, for 0.5 mol/L Sn(II)2-ethylhexanoate in xylene (◊), burning in 0.4 L/min
coflowing oxygen. The initial velocity is 𝒗𝟎 = 𝟎. 𝟓𝟕𝒎/𝒔
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Thermogravimetric Analysis (TGA) of Sn(II)-, Cr(III)- and Ce(III)2-ethylhexanoate
Figure B: TGA of Sn(II)2-ethylhexanoate, Cr(III)2-ethylhexanoate and Ce(III)2-ethylhexanoate, respectively, for a heating rate of
7.5°K/min from room temperature to 1200°C in air. The initial (mini) and residual (mres) masses are 63.5mg, 51.3mg, 66.3mg and
34.3mg, 28.5mg, 28.9mg respectively.
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