S udy and cha ac e iza ion o he ins abili ies
gene a ed in expanding sphe ical lames o
hyd ogen/me hane/ai mix u es
M. Reyes
a,*
, R. Sas e
a
, F.V. Tinau
b
, J. Rod ı
´guez-Fe n
andez
c
a
Depa men o Ene gy and Fluid Mechanics Enginee ing, Uni e si y o Valladolid, Paseo del Cauce s/n, 47011,
Valladolid, Spain
b
CMT-Mo o es T
e micos, Uni e si a Poli 
ecnica de Val
encia, 46022, Valencia, Spain
c
Uni e sidad de Cas illa la Mancha. ETS Ingenie ı´a Indus ial (Edi icio Poli 
ecnico), A da. Camilo Jos
e Cela,3, 10371,
Ciudad Real, Spain
highligh s
O igin and na u e o ins abili ies de eloped in sphe ical hyd ogen/me hane lames.
Di e en ia e ins abili ies o hyd odynamic and he mal-di usi e o igin.
Ins abili y peninsulas calcula ed o es ablish s abili y limi s in each mix u e.
Lewis e ec i e numbe de e mines he shape and loca ion o ins abili y peninsulas.
a icle in o
A icle his o y:
Recei ed 11 Ma ch 2022
Recei ed in e ised o m
3 May 2022
Accep ed 8 May 2022
A ailable online xxx
Keywo ds:
Combus ion ins abili ies
The mal-di usi e and
hyd odynamic e ec
Peninsula o ins abili y
Cellula i y
G ow h a e o ins abili ies
Schlie en pho og aphy
abs ac
In he p esen wo k an analysis o he o igin and na u e o in insic ins abili ies in com-
bus ion p ocesses wi h di e en hyd ogen/me hane mix u es is de eloped. These
expanding sphe ical lame on expe imen s ha e been de eloped in a cylind ical con-
s an olume combus ion bomb, which allows eco ding he p ocess h ough Schlie en
pho og aphy me hod.
The s abili y s udy in combus ion p ocesses has a g ea impo ance o assu e hei
secu i y and con ol, since he unde s anding o lame ins abili ies is necessa y o
imp o ing he in e nal combus ion engines pe o mance.
To ca y ou his men ioned s udy, a e iew o he concep s and pa ame e s used in
sphe ical lame on ins abili ies esea ch is i s p oposed, as well as a physical expla-
na ion o each concep and he ela ions among hem.
Addi ionally, a me hodology ha aims o de e mine he in luence o he uel mix u es
in he o igin and de elopmen o he lame on ins abili ies is sugges ed. Mo eo e , he
in insic e ec s o he combus ion p ocess, such as he he mal-di usi e and he hyd o-
dynamic e ec , a e sepa a ely s udied, including hei indi idual con ibu ions o he
g ow h a e o ins abili ies which allows o de e mine combus ion na u e and o ob ain he
ins abili y peninsula o each uel mix u e.
*Co esponding au ho .
E-mail add ess: [email p o ec ed] (M. Reyes).
A ailable online a www.sciencedi ec .com
ScienceDi ec
jou nal homepage: www.else ie .com/loca e/he
in e na ional jou nal o hyd ogen ene gy xxx (xxxx) xxx
h ps://doi.o g/10.1016/j.ijhydene.2022.05.063
0360-3199/©2022 The Au ho (s). Published by Else ie L d on behal o Hyd ogen Ene gy Publica ions LLC. This is an open access a icle unde he CC BY-NC-
ND license (h p://c ea i ecommons.o g/licenses/by-nc-nd/4.0/).
Please ci e his a icle as: Reyes M e al., S udy and cha ac e iza ion o he ins abili ies gene a ed in expanding sphe ical lames o
hyd ogen/me hane/ai mix u es, In e na ional Jou nal o Hyd ogen Ene gy, h ps://doi.o g/10.1016/j.ijhydene.2022.05.063
Finally, his me hodology includes a quali a i e s udy o he cellula i y phenomenon
(when he ins abili ies de elop all o e he lame on ), conside ing he pa ame e s which
in luence on his phenomenon.
©2022 The Au ho (s). Published by Else ie L d on behal o Hyd ogen Ene gy Publica ions
LLC. This is an open access a icle unde he CC BY-NC-ND license (h p://
c ea i ecommons.o g/licenses/by-nc-nd/4.0/).
In oduc ion
Nowadays, he e is an ene gy p oblem wo ldwide based on
he one hand, on a p ac ically exclusi e dependence on ossil
uels de i ed om pe oleum, and, on he o he hand, on he
e ec ha hei use has on he en i onmen due o he
emissions gene a ed in hei combus ion p ocess. I is es i-
ma ed ha he anspo sec o is esponsible o almos 30%
o o al CO
2
emissions in he Eu opean Union (EU), wi h his
gas being he main cause o he g eenhouse e ec . A possible
solu ion is he eplacemen o hese CO
2
-p oducing uels wi h
o he s wi h low o no ca bon con en such as hyd ogen (H
2
)o
ammonia (NH
3
), being mo e sus ainable om he en i on-
men al poin o iew.
P io o he in oduc ion o any uel mix u e in a con en-
ional engine, i s beha io mus be s udied by isola ing i om
o he ex e nal ac o s ha may a ec he pe o mance o
combus ion, such as he u bulence ha occu s in an engine
due o he mo emen o he pis ons. Cons an olume com-
bus ion bombs (CVCB) allow he analysis o mix u es in
lamina egime, hus acili a ing he s udy o hei na u e. The
abo e app oach is a he base o his esea ch wo k, in which
lamina combus ion o sphe ical lames o mix u es o H
2
/
CH
4
/ai wi h di e en pe cen ages o hyd ogen (H
2
) a e cha -
ac e ized in a CVCB wi h cylind ical geome y.
A undamen al aspec in he cha ac e iza ion o uel
combus ion is he s udy o i s s abili y, which consis s o he
analysis o he de elopmen o he ins abili ies inhe en o he
combus ion p ocess: o igin, na u e ( he mo-di usi e o hy-
d odynamic, mainly), g ow h, e c. These can al e he
mo phology o he lame and a ec he bu ning eloci y.
The e is a g owing in e es in unde s anding and con olling
he uns able beha io o cellula lames [1] and he ansi ion
o cellula i y in expanding sphe ical lames [2e4].
The e a e se e al ways o cha ac e ize a combus ion p o-
cess depending on he me hod and esou ces used, bu all o
hem aim o de e mine he mos signi ican pa ame e s ha
de ine how a combus ion is de eloping: he modynamic
condi ions, bu ning eloci y, ins abili ies, and lame
mo phology among o he s. The Schlie en echnique is used in
his wo k o eco d he lame on de elopmen . This ech-
nique is based on he ligh de lec ion caused by densi y g a-
dien s [5]. I was i s employed by Toeple in 1864 [6]. The
Schlie en me hod is widely used o he s udy o lames in a
combus ion p ocess since i allows hei eco ding wi h a g ea
cla i y [7]. Thanks o he use o his echnique, i is possible o
s udy he mo phology [8,9], ins abili ies and cellula i y o he
lame on , as well as i s bu ning eloci y [4,5,8,10] and,
he e o e, cha ac e ize he uel mix u e conside ed [11,12].
Spon aneous pe u ba ions appea in lames due o a ious
phenomena such as acous ic ib a ions o p essu e wa es [4].
In insic combus ion e ec s cause ha he ampli ude o he
wa es on he lame on dec eases o inc eases depending on
he dominan ac ion o hose e ec s. The esul o each e ec
in he lame on is he ins abili y, independen ly o hei
s abilizing o des abilizing cha ac e [13,14]. The phenomena
ha can p oduce ins abili ies in he lame on (in a lamina
egime) o a p emixed combus ion a e h ee: olume o ces,
hyd odynamic e ec s and he mo-di usi e e ec s [15].
Volume o ces a e ele an when a less dense luid mo es
owa ds a dense one in he opposi e di ec ion o ha o a
olume o ce ( o example, g a i y). Then, he e appea s an
ins abili y due o olume o ces (also known as Rayleigh-
Taylo [16]). In a lame, his occu s when he combus ion
p oduc s (wi h lowe densi y) ad ance owa ds he esh
mix u e (highe densi y) upwa ds (con a y o g a i y); he
lame on is hen conside ed as a discon inui y o densi ies.
This ins abili y ype, compa ed wi h he o he wo, has he
lowe in luence on he de elopmen o he lame.
The second ins abili y conside ed is he hyd odynamic one
(Da ieus-Landau, DL), which is associa ed wi h he he mal
expansion o densi y di e ence be ween he unbu n and
bu n gases in a combus ion p ocess, delimi ed by he lame
on [13,14]. The hyd odynamic e ec gene a es des abilizing
ins abili ies since hey end o in ensi y all o he lame on
pe u ba ions signi ican ly la ge han he lame on hick-
ness (d
l
)[15,17]. Hyd odynamic ins abili y can also be s imu-
la ed by educed lame hickness and high p essu es [18]. In
addi ion, u bulen p opaga ion speed can be enhanced by he
onse o he hyd odynamic ins abili y [19,20].
The he mal-di usi e e ec is he cause o he hi d
ins abili y ype ha a ec s he combus ion p ocess in a sig-
ni ican way. This ins abili y does no ha e a unique s able o
uns able cha ac e since i in ol es he mal and molecula
di usi i ies, ha ha e opposi e esul s [15,21e24]. The mo-
lecula di usi i y a ec s he pe u ba ions in such a way ha
i inc eases hem (des abilizing con ibu ion); on he con a y,
he he mal di usi i y ends o a enua e he lame on
(s abilizing con ibu ion). This e ec can be ep esen ed
quan i a i ely by he dimensionless Lewis numbe (Le), which
is he a io be ween he mal di usi i y (a) and molecula
di usi i y (D).
Des abilizing he mal-di usi e ins abili y occu s when he
Le is smalle han a c i ical alue (when he molecula di u-
si i y domina es he p ocess), and c acks and cells de elop
in e na ional jou nal o hyd ogen ene gy xxx (xxxx) xxx2
Please ci e his a icle as: Reyes M e al., S udy and cha ac e iza ion o he ins abili ies gene a ed in expanding sphe ical lames o
hyd ogen/me hane/ai mix u es, In e na ional Jou nal o Hyd ogen Ene gy, h ps://doi.o g/10.1016/j.ijhydene.2022.05.063
om he beginning on he lame on su ace; o he wise, he
lame should be unw inkled a i s , al hough a e a ce ain
ime, needed o he hyd odynamic ins abili y o appea ,
w inkles can be isible [25]. In addi ion o he la e , he in-
luence o he lame s e ch mus be conside ed in sphe ical
lames. This pa ame e includes s ain and cu a u e e ec s
due o he lame geome y and always ends o a enua e he
ins abili ies in he combus ion beginning (in insically s able
cha ac e ). I should be s udied oge he wi h he na u e o he
mix u e and, speci ically, he he mo-di usi e e ec [24,26]
ep esen ed by he Lewis numbe (Le) o be e , he e ec i e
Lewis numbe (Le
e
), see Re s. [21,24,27,28], which is a
weigh ed a e age o he eac an s Lewis numbe s.
The c i ical adius and he cellula adius a e s udied in
li e a u e o cha ac e ize he onse o ins abili y on lames and
he cellula phenomenon, espec i ely [24,29e33]. The
dimensionless lame adius is he Pecle numbe (Pe), which is
he a io be ween he ins an aneous adius and he lame
hickness. Many de ini ions o c i ical adius can be ob ained
in li e a u e, B adley e al. [34] de ined he c i ical adius a he
ins an o he appea ance o he i s c acks in he lame on .
Bech old and Ma alon [2] iden i ied he c i ical adius, o
lames wi h e ec i e Lewis numbe bigge han a c i ical
alue, as he adius beyond which he lame is uns able due o
he dominance o he hyd odynamic e ec when ins abili ies
a e no longe dampened, al hough isually i is no app e-
ciable. This is he de ini ion o c i ical adius adop ed in his
wo k, associa ed wi h he c i ical Pecle numbe (Pe
c
). Ano he
de ini ion e e s o he lame adius when he lame is isibly
uns able due o he de elopmen o cellula i y. This adius can
be called c i ical cellula adius, also u ilized by B adley e al.
[34]. F om his adius, he cellula i y inc eases he lame on
su ace, which p oduces an augmen a ion in he appa en
combus ion bu ning eloci y (o be e , combus ion a e). In
he wo ks o Jiang e al. [35] and Tinau e al. [5] his de ini ion
o he c i ical adius is applied.
The unde s anding o hyd ogen/me hane combus ion and
lame ins abili ies is necessa y o imp o ing in e nal com-
bus ion engines pe o mance. P e ious wo ks ha e been
ocused on he bu ning eloci y s udy o hyd ogen/me hane
mix u es [36e42]. O he s udies ha e ocused on he engine
combus ion [43e45]. Va ious esea che s ha e in es iga ed
he cell o ma ion in hyd ogen/ai lames and in mix u es o
hyd ogen wi h o he hyd oca bons lames [46e49], indica ing
ha hyd ogen addi ion inc eased hyd odynamic and he mal-
di usi e ins abili ies. Addi ionally, an unde s anding o he
o ma ion and o igin o cellula ins abili ies in hyd ogen/
me hane-ai lames should be conside ed u he . Jiang e al.
[50] s udied he cellula s uc u e o me hane/hyd ogen/ai
lames in a sphe ical combus o , o a ious equi alence a-
ios and hyd ogen ac ions, and s udied a co ela ion be-
ween he cellula s uc u e and p essu e. Kim e al. [51]
s udied cellula ins abili ies on expanding sphe ical p opa-
ga ion hyd ogen/ai e, me hane/ai and p opane/ai lames.
They cha ac e ized he size o he cell, obse ing bigge cells
due o hyd odynamic ins abili y han he gene a ed by he
he mal-di usi e. The c i ical Pecle numbe is also a ec ed
by he ini ial p essu e, dec easing wi h he inc emen o
p essu e. Zhang e al. [52] made a lame dynamics analysis o
na u al gas en iched wi h hyd ogen in p emixed u bulen
combus ion, ob aining an inc emen o he w inkled s uc u e
wi h he u bulence in ensi y and wi h he hyd ogen a io.
Cellula ins abili ies enhance he global p opaga ion a e o
lamina lames and ha e also been s udied by many e-
sea che s [30,47,53e55]. Kwon e al. [56] s udied hyd ogen
lames a ele a ed p essu es o iden i y he e ec s o he mal
expansion a io, lame hickness, Lewis numbe , and s e ch
a e on he gene a ion o hyd odynamic and di usional-
he mal ins abili ies in he lame on . Jiang e al. [50] in es-
iga ed cell o ma ion in hyd ogen/me hane p emixed lames
and ob ained a co ela ion be ween cellula s uc u e o he
lame on and p essu e. Sun e al. [30] s udied he cellula
ins abili ies o hyd ogen-ai p emixed lames a di e en
ini ial condi ions, showing ha , o lean hyd ogen/ai lames,
he cellula ins abili ies a e domina ed by he he mal-
di usi e ins abili y, and o s oichiome ic and ich
hyd ogen/ai lames, he cellula s uc u e is in luenced by he
hyd odynamic ins abili y. Oka o e al. [57] ha e de eloped an
expe imen al and heo e ical in es iga ion o cellula in-
s abili ies in lean hyd ogen/me hane mix u es in a cons an
olume combus ion chambe , ob aining an inc emen o he
cellula ins abili y and sel -accele a ion wi h he inc emen o
he hyd ogen con en and mix u e p essu e. Hu e al. [47]
obse ed ha he inc emen o hyd ogen in me hane/
hyd ogen/ai mix u es p omo es cellula i y in lean lames.
Law e al. [58] s udied he ou wa d p opaga ion and de elop-
men o su ace cellula ins abili y o spa k-igni ed sphe ical
p emixed lames o mix u es o hyd ogen, hyd oca bon, and
ai , showing ha p opane subs i u ion mode a es cell o -
ma ion due o bo h di usionale he mal as well hyd ody-
namic ins abili ies. Smallbone e al. [59] ca ied ou s udies o
lame ins abili y, lamina and u bulen eloci y o hyd ogen/
ai mix u es in a an s i ed combus ion bomb. Wang e al.
[60,61] in es iga ed he lame on s uc u e o syngas p e-
mixed lames. Vu e al. [1] in es iga e he cell o ma ion in
hyd ogen/me hane/monoxide eai p emixed lames. Jiang
e al. [50] s udied he in luence o p essu e in he de elopmen
o a cellula s uc u e.
In p esen wo k an analysis o he o igin and na u e o
in insic ins abili ies in sphe ical expanding lames o
hyd ogen/me hane blends is de eloped, because he e is a
lack o da a in li e a u e. These sphe ical lame on expe i-
men s ha e been pe o med in a cylind ical cons an olume
combus ion bomb (CylCVCB), which allows image eco ding
h ough Schlie en pho og aphy me hod and eco ding he
ins an aneous p essu e. Wi h he me hodology p esen ed in
his esea ch, i is possible o ob ain he s abili y maps o
di e en combus ion mix u es, which co esponds wi h he
images di ec ly ob ained. To ca y ou his esea ch, a e iew
o he concep s and pa ame e s used in he s udy o in-
s abili ies in sphe ical lames is i s p oposed, as well as a
physical explana ion o each concep and he ela ions among
hem. A e ha , he expe imen al ins alla ion and me hod-
ology a e explained, and inally expe imen al esul s a e p e-
sen ed o mix u es o me hane/hyd ogen wi h di e en
pe cen age o hyd ogen ( om 0 o 100%), o wo di e en
uel/ai equi alence a ios, and ini ial p essu e and empe a-
u e cons an in all he expe imen s.
in e na ional jou nal o hyd ogen ene gy xxx (xxxx) xxx 3
Please ci e his a icle as: Reyes M e al., S udy and cha ac e iza ion o he ins abili ies gene a ed in expanding sphe ical lames o
hyd ogen/me hane/ai mix u es, In e na ional Jou nal o Hyd ogen Ene gy, h ps://doi.o g/10.1016/j.ijhydene.2022.05.063
S abili y analysis in sphe ical lames
To cha ac e ize an uns able sphe ical lame, i is impo an o
di e en ia e be ween he ins an a which ins abili ies a e
o igina ed (a he i s s ages o he combus ion o in he
subsequen ins an s o he de elopmen o he combus ion
p ocess) and whe he o no he e olu ion o he ins abili y
leads o he de elopmen o a cellula lame on .
The g ow h a e o sphe ical lame ins abili ies is de ined
as he inc ease in ime o he ela i e ampli ude o he
pe u ba ion ( a io be ween he ampli ude and he ins an a-
neous adius o he sphe e) o med on he su ace o he lame
[14] as he join e ec o he hyd odynamic and he mal-
di usi e ins abili ies.
When he lame adius has eached a ce ain alue
0
(signi ican ly g ea e han he hickness o he lame, dl) he
ampli ude Ao he pe u ba ion can be non-dimensionalised
wi h o gi e a. Once pe u bed a iables a e expanded in a
sphe ical ha monic se ies, a akes he ollowing o m (adap ed
om Re . [4] o cohe ence wi h he subsequen exp essions
o he e ms appea ing in i ):
a¼a0RuþdlU
,ln R(1)
whe e a0is he ini ial dimensionless ampli ude when ¼
0
;
R¼ /
0
is he dimensionless lame adius; dlis he lame on
hickness (dl¼a=Sl, whe e ais he he mal di usi i y o he
mix u e and Sl he lamina eloci y [10]). uis associa ed wi h
hyd odynamic ins abili y [4,21,29,32]; i is a dimensionless
pa ame e (usually deno ed as uDL in i s equi alen in la
lames), ha only depends on he he mal expansion sand he
dimensionless wa enumbe n.Uis associa ed wi h he mal-
di usi e ins abili y [29]; i is also dimensionless and de-
pends on he he mal expansion, Lewis numbe and Zeldo ich
numbe (b) (see below). The dimensionless wa enumbe n
quan i ies he numbe o wa es, wi h a lwa eleng h, ha i
in a ci cum e ence o adius n¼2p
l, as illus a ed in Fig. 1.
The g ow h a e o an ins abili y is
1
=aðda =d Þ[4,21], he e-
ina e iden i ied by he le e S, will adop di e en o ms
depending on he case. The gene al unc ion o Sð Þis aken
om he esea ch o Addabbo, Bech old and Ma alon [21,32],
collec ed in he s udy o Lapalme e al. [24] and ep esen ed in
equa ion (2).
Sð Þ¼1
a
da
d ¼1
d
d udl
U¼1
d
d uU
Pe(2)
In equa ion (2) i is possible o see he con ibu ions o he
ins abili y g own a e due o he hyd odynamic (u) and
he mo-di usi e ( U
Pe) e ec s [29]. Hyd odynamic e ec is
always posi i e (con ibu ing o ins abili y), bu he he mo-
di usi e e ec can inc ease o educe he i s , depending
on he alue o U:
U¼Q1þbLee 1
s1Q2þP Q3(3)
No e ha Uconsis s o h ee con ibu ions, each accom-
panied by a Qicoe icien . The physical meaning o each o
hem is as ollows: he e m Q1b ings he in luence o he
he mal conduc i i y (s able cha ac e ); he second e m
bLee 1
s1Q2 ep esen s he e ec o he molecula di usion
(s able o uns able) and he e m P Q
3
(P is he P and l num-
be ) b ings he e ec o he iscosi y (s able) [62]. The co-
e icien s Q
1
,Q
2
and Q
3
a e gi en in Eqs. (4)e(6).
Q1¼g1
sDn4ðsþ1Þþsn3ð2uþ5Þþn2us 2s2þs1
þnsðs73usuÞ2sð1þuÞ(4)
Q2¼g2ðs1Þ
2D2n4þn3ð2us þ2uþ10s3Þþn22su2þð5s1Þu
þ3s2s22þnsu2ð14sÞ14s2þ1uþ39s8s2
2su2þ4uþ3 (5)
Q3¼2nðn21Þðs1Þ
sD½ðnþ2Þð
~
kðxÞg3Þ3ð
~
lðxÞ1Þ (6)
whe e D¼2auþb2a. The de ini ions o coe icien s g1;g2
and g3appea in he second column o Table 1. I can be seen
ha he h ee gidepend on he expansion a io sand a
unc ion
~
kðxÞwhich ep esen s he he mal conduc i i y
exp essed in e ms o he dimensionless empe a u e T=Tu
(i.e., he lame empe a u e To e he unbu ned gas empe -
a u e Tu) and scaled by i s alue in he unbu ned gas. In o de
o ob ain explici exp essions o gi ha s ill e ain he main
dependence on s, di e en possibili ies o he a ia ion o
~
kðxÞ
can be conside ed: cons an (o de o 1), a iable as ffiffiffi
T
p, and
a iable as T. The exp essions o gi o he h ee possible de-
pendences o
~
kðxÞappea in he addi ional columns o Table 1.
In he i s case (
~
k(x)¼1), wi h he hypo hesis o cons an
anspo p ope ies, he mal conduc i i y is a cons an and
he in luence o iscosi y on he s abili y o he lame is min-
imal [21]. I is wo h men ioning he o mula ion made by
B adley and Ha pe in 1994 [4] wi h he assump ion o con-
s an p ope ies. Coe icien s Q
i
a e also de ailed in B adley [3]
which desc ibes ou s ages in he p opaga ion o a lamina
sphe ical lame, hese a e: (1) s able lamina p opaga ion, (2)
c acking o he lame on and cell o ma ion, (3) p opaga ion
o he cellula lame, (4) u bulen p opaga ion.
Fig. 1 eFlame adius, wa e ampli ude and wa eleng h.
in e na ional jou nal o hyd ogen ene gy xxx (xxxx) xxx4
Please ci e his a icle as: Reyes M e al., S udy and cha ac e iza ion o he ins abili ies gene a ed in expanding sphe ical lames o
hyd ogen/me hane/ai mix u es, In e na ional Jou nal o Hyd ogen Ene gy, h ps://doi.o g/10.1016/j.ijhydene.2022.05.063
The e ec o he s e ch canno be s udied sepa a ely
because he na u e o he mix u e mus be conside ed and,
speci ically, he he mo-di usi e e ec e lec ed in he a io
be ween he e ec i e Lewis numbe Le
e
in a mix u e, and he
c i ical Lewis numbe o he combus ion (Le
e *
). Equa ion (7)
shows his la e pa ame e , whe e bis he Zeldo ich num-
be ; as o Q
1
and Q
3
, hey mus be e alua ed o he c i ical
adius o Pecle numbe acco ding o equa ions (4) and (6).
Lee *¼1s1
bQ2ðQ1þQ3P Þ(7)
As i was explained be o e, i in a uel mix u e Le
e
<Le
e *
,i
means ha he he mo-di usi e e ec is des abilizing, and
ins abili ies g ow om he beginning. On he con a y, in
combus ions wi h Le
e
>Le
e *
he he mo-di usi e e ec is
s abilizing and, he e o e, he only e ec capable o inc easing
ins abili ies is he hyd odynamic one.
I is possible o ob ain he cu es o he g ow h a e o he
ins abili y Swhen he alues o Eq. (2) a e ep esen ed e sus
he wa enumbe no al e na i ely he wa eleng h o he
pe u ba ions l o a pa icula lame on adius o ime
[2,4,64], as can be seen in Fig. 2, whe e i is possible o see wo
egions: s able and uns able. Bo h zones a e sepa a ed by he
alue o S¼0, he s abili y limi o a sphe ical lame. The
do ed line co esponds wi h he c i ical adius
c
om which
he e a e ce ain wa eleng hs whose g ow h a e is posi i e
(uns able lame).
Expe imen al appa a us and p ocedu e
Expe imen al se up
In his esea ch wo k, expe imen s we e conduc ed in a cy-
lind ical cons an olume combus ion essel (CylCVCB) [5],
wi h he ollowing dimensions: diame e 114 mm and heigh
135 mm, o in es iga e he combus ion de elopmen and in
pa icula , ins abili ies and he onse o cells in he lame
su ace, see Fig. 3. The combus ion chambe was designed
wi h wo op ical windows made o used silica on he side-
walls o he cylinde . The ins alla ion is ins umen ed o
measu e ins an aneous p essu e du ing he combus ion
(piezoelec ic p essu e ansduce Kis le ype 7063) and o
egis e he lame de elopmen wi h a high-speed Schlie en
pho og aphy sys em: lame images we e eco ded using a
high-speed came a Phan om V210 a 7000 ames pe second
( esolu ion 832 800 and exposu e ime o 10 ms).
Hyd ogen, me hane and ai a e indi idually in oduced
om p essu e anks using a pa ial p essu e me hod, and an
igni ion sys em s a s he combus ion a he cen e o he
combus ion chambe ( o mo e in o ma ion abou he expe -
imen al ins alla ion see Re . [5]).
An algo i hm is used o p ocess he images ob ained wi h
he Schlie en echnique (high-speed came a), o ob ain he
ime e olu ion o he lame adius and o s udy he cellula
s uc u e which appea s as a w inkling o he lame su ace.
Radius e olu ion is ob ained h ough ame by ame p o-
cessing, a e backg ound emo al, bina iza ion and hinning
ans o ma ion o highligh he bo de s be ween cells, as i is
explained in de ail in Re . [5], ob aining he ins an aneous
posi ion o he lame on (wi h a andom sample consensus
algo i hm p og ammed in Ma lab) and iden i ying he cells
ha appea in he lame du ing de combus ion de elopmen .
Ins an aneous p essu e is egis e ed inside he combus ion
chambe , and i is analyzed by means o a wo-zone diagnosis
model, wi h he aim o ob aining bu ning eloci y, Lewis
numbe and e ec i e Lewis numbe (Le and Le
e
), he mal
expansion s, ins abili y g ow h a e S, and he es o he pa-
ame e s necessa y o s udy he ins abili y g own a e and
hei coe icien s, explained in de ail in p e ious sec ions.
Table 1 eAnaly ical exp essions ha de ine he coe icien s g
i
and p ac ical as a unc ion o he he mal conduc i i y
[21,63].
~
kðxÞ¼1
~
kðxÞ¼ ffiffiffiffi
T
p~
kðxÞ¼T
g1s
s1Z
s
1
~
kðxÞ
xdx
s
s1ln s2s
ffiffiffi
s
pþ1
s
g21
s1Z
s
1
~
~
kðxÞ
xlns1
x1dx 1
s1Z0
∞
ln½1þðs1Þezdz
4
s1 ffiffiffi
s
p1ln½0:5ðffiffiffi
s
pþ1Þg 1
g31
s1Z
s
1
~
kðxÞdx
12ðs3=21Þ
3ðs1Þ
sþ1
2
Fig. 2 eIns abili y g ow h a e cu es o a gi en lame
adius as a unc ion o he pe u ba ion wa eleng h.
in e na ional jou nal o hyd ogen ene gy xxx (xxxx) xxx 5
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hyd ogen/me hane/ai mix u es, In e na ional Jou nal o Hyd ogen Ene gy, h ps://doi.o g/10.1016/j.ijhydene.2022.05.063

Me hodology
In his esea ch wo k, mix u es o H
2
/CH
4
wi h di e en pe -
cen ages o hyd ogen ( a ying om 0, 20, 50, 80 and 100%) a e
analyzed o de e mine he in luence o hyd ogen addi ion on
he s abili y o a lame, speci ically on he o igin and na u e o
he ins abili ies in insic o a combus ion p ocess. To isola e
he e ec o hyd ogen om o he possible in luencing a i-
ables, wo g oups o expe imen s wi h s oichiome ic and lean
uel/ai a io (F¼1 and F¼0.7) ha e been selec ed; wi h an
ini ial p essu e o 0.1 MPa, and an ini ial empe a u e o 323 K.
The i s g oup (F¼1) has been s udied o iden i y he in lu-
ence o he hyd ogen in he lame s abili y analysis (s abili y
limi s, combus ion na u e, ins abili ies, and cellula i y). The
aim o he second g oup is o s udy he impac o lean
equi alence a io on he s abili y analysis, as well as he e ec
o he hyd ogen con en , by compa ing he esul s wi h hose
o s oichiome ic equi alence a io.
The s abili y s udy in lames o H
2
/CH
4
eai mix u es is
made by examining ou undamen al cha ac e is ics o a
sphe ical lame: s abili y limi s ela ed o pe u ba ions
wa eleng h l, lame na u e, ins abili ies g ow h a e Sand he
cellula i y phenomenon. In his sec ion, i is b ie ly explained
he meaning o each one o hem and hei main goals. These
ou cha ac e is ics a e de eloped and applied o he selec ed
expe imen s in he sec ion co esponding o esul s.
S abili y limi s ela ed o pe u ba ions wa eleng h (l)
To s udy he s abili y limi s ela ed o he wa eleng h o
he pe u ba ions (l) i is use ul he g aphical ep esen a ion
o he s abili y limi s o a lame, which is called s abili y cu e
o ins abili y peninsula [13]. Fig. 4i shows an example o his
cha whe e he wa enumbe s o which ins abili ies g ow h
a e is ze o (S¼0) a e plo ed e sus he Pecle numbe .
Simila ly, ins ead o he Pecle numbe , he lame adius could
also be u ilized ( his is employed in he esul s sec ion o his
pape ).
These alues con o m a egion ha con ains he wa e-
numbe s o all he pe u ba ions ha expe ience an inc e-
men o hei ampli ude a a ce ain lame Pecle numbe (o
adius) since hei g ow h a e is posi i e (S>0); he e o e, i is
called he uns able egion ( he blue one). The o he a ea is he
s able egion, in which he g ow h a e o he pe u ba ions is
nega i e (S<0) and consequen ly he wa es a e a enua ed. I
should be no ed ha he nose o he peninsula co esponds o
he c i ical Pecle numbe (o c i ical lame adius) om which
he lame en e s in he uns able egion. Fo a Pecle numbe
la ge han he c i ical alue, he e is a se ies o g owing
pe u ba ions de ined by hei wa enumbe s (wa eleng hs)
be ween n
max
(o l
min
) and n
min
(o l
max
), while all hose which
a e ou side his ange a e dec easing. Fig. 4ii shows a g ow h
a e cu e o a gi en Pecle numbe in which hese limi s a e
ma ked. These maximum and minimum alues ul ill he
ollowing condi ion o each Pecle numbe (and associa ed
lame adius):
S¼0¼1
a
da
d ¼1
d
d udl
U0uU
Pe ¼00Pe ¼U
u(8)
Fig. 4iii con ains a join ep esen a ion o he in o ma ion
o he ins abili y g ow h a e cu e o a gi en Pecle numbe
and he ins abili y peninsula.
Fig. 3 eSchema ics o he expe imen al se up.
in e na ional jou nal o hyd ogen ene gy xxx (xxxx) xxx6
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hyd ogen/me hane/ai mix u es, In e na ional Jou nal o Hyd ogen Ene gy, h ps://doi.o g/10.1016/j.ijhydene.2022.05.063
Flame s abili y na u e.
A combus ion is conside ed in insically uns able when
he lame on ins abili ies appea and g ow al eady om he
beginning o he p ocess, i.e., i he ins abili y g ow h a e is
posi i e (S>0) a he i s s a es o he combus ion. On he
con a y, i he ins abili ies begin o g ow only a e he lame
adius has eached a ce ain alue (e en i i is small), a
combus ion is conside ed s able. The e o e, he combus ion
s abili y cha ac e is de ined by he ins abili ies g ow h a e in
he i s s ages o he p ocess. As men ioned be o e, he hy-
d odynamic e ec has always a des abilizing cha ac e .
Howe e , he he mal-di usi e e ec could be s abilizing
(coun e ac ing he hyd odynamic e ec oge he wi h he
lame s e ch) o des abilizing (inc easing he hyd odynamic
e ec ), meaning ha he combus ion s abili y na u e is
s ongly dependen on his second e ec .
This sec ion aims o cha ac e ize he combus ion na u e
and he mo phology o a s abili y cu e de i ed om an un-
s able combus ion.
Wi h he objec i e o iden i ying he s abili y na u e o a
lame, he alue o i s e ec i e Lewis numbe mus be
compa ed wi h he alue o i s c i ical e ec i e Lewis numbe .
I he e ec i e Lewis numbe alue is g ea e han he c i ical
one (Le
e
>Le
e *
) he combus ion is s able a he beginning, i s
ins abili y peninsula is loca ed o he igh o he e ical axis,
he ins abili ies ha e a hyd odynamic sou ce and s a o g ow
only once he Pecle numbe has a alue highe han 0.
O he wise, when Le
e
;<Le
e *
, he combus ion has an uns able
na u e, he ins abili ies a ise om he he mal-di usi e e ec
al eady om he beginning (in addi ion o he hyd odynamic
con ibu ion.
Following he p e ious app oach, he limi be ween he
s able and uns able combus ion is well ma ked by he e ec-
i e Lewis numbe whose c i ical alue is, in gene al, close o
1. In Fig. 5 wo ins abili y peninsulas o di e en Lewis
e ec i e numbe s a e p esen ed. When he Lewis numbe is
equal o i s c i ical alue (Le
e
¼Le
e *
), he peninsula nose
ouches he e ical axis, i.e., he c i ical Pecle numbe o he
combus ion is equal o ze o (b oken line in Fig. 5). When he
Lewis numbe is lowe han i s c i ical alue (Le
e
>Le
e *
), he
peninsula is displaced o he igh in he igu e.
Ins abili ies g ow h a e and ela i e con ibu ions o hy-
d odynamic and he mal-di usi e e ec s.
As p e iously men ioned, he in luences o hyd odynamic
and he mal-di usi e con ibu ions a e e lec ed on he in-
s abili ies g ow h a e (see Eq. (2) and Eq. (3)). The he mal-
di usi e e ec has mo e in luence a he p ima y s ages o
he combus ion p ocess, since i is in e sely p opo ional o
he Pecle numbe ( U
Pe). When he lame adius inc eases,
his e ec loses impo ance, and he g ow h a e is domina ed
by he hyd odynamic e ec (u). Consequen ly, he mo e s a-
bilizing powe he he mal-di usi e con ibu ion has, he
la e he ins abili ies due o he hyd odynamic con ibu ion
appea . Acco ding o he abo e, ega dless o he combus ion
Fig. 4 eSchemes o he s abili y cu e.
Fig. 5 eIns abili y peninsulas o wo mix u es wi h
di e en alues o Le
e
.
in e na ional jou nal o hyd ogen ene gy xxx (xxxx) xxx 7
Please ci e his a icle as: Reyes M e al., S udy and cha ac e iza ion o he ins abili ies gene a ed in expanding sphe ical lames o
hyd ogen/me hane/ai mix u es, In e na ional Jou nal o Hyd ogen Ene gy, h ps://doi.o g/10.1016/j.ijhydene.2022.05.063
na u e, all he lames would isibly de elop ins abili ies in a
combus ion chambe wi h an in ini e size.
Cellula i y de elopmen .
The cellula i y phenomenon is he isible consequence o
he ull de elopmen o he ins abili ies in a lame on . A a
ce ain adius, he wa es ampli udes a e so big ha hey
ins an ly b eak down he lame on s uc u e in such a way
ha cells a e sp ead all o e he lame su ace in a pe cep ible
way, which can be eco ded by he pho og aphic schlie en
echnique. A quali a i e analysis o he lames can be ca ied
ou h ough he s udy o hose images in o de o es ablish he
mo e- and less-de e minan pa ame e s on he appa i ion
and de elopmen o he cellula i y phenomenon.
Resul s o he s abili y o hyd ogen/me hane
lames
In his sec ion esul s o he s abili y o hyd ogen-me hane
lames a e p esen ed o di e en pe cen ages o hyd ogen
in he lame mix u e and o wo di e en uel/ai equi alence
a ios: s oichiome ic and lean mix u e (F¼0.7).
Fo all cases, he alues o he expansion a io sa e
calcula ed o each ime. The alues o each mix u e e ec i e
Fig. 6 eIns abili y peninsulas.
in e na ional jou nal o hyd ogen ene gy xxx (xxxx) xxx8
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hyd ogen/me hane/ai mix u es, In e na ional Jou nal o Hyd ogen Ene gy, h ps://doi.o g/10.1016/j.ijhydene.2022.05.063
Lewis numbe a e he co esponding o he c i ical adius o
he s able ones and o he ini ial adius o hose uns able,
always in he unbu ned side. The c i ical Lewis numbe is
calcula ed acco ding o Eq. (7).
S abili y limi s
In Fig. 6 he s abili y maps e sus he Pecle numbe (le ) and
he lame adius ( igh ) a e p esen ed o all he expe imen s
de eloped. Compa ing hese wo ep esen a ions, i can be
emo ed he lame hickness in luence. In Fig. 6i and 6ii s a-
bili y cu es a e p esen ed o he expe imen s de eloped a
s oichiome ic uel/ai equi alence a io o di e en
hyd ogen con en (0%, 20%, 50%, 80% and 100%). I can be seen
ha bo h peninsulas o he mix u es wi h 50% H
2
o F¼1a e
a he le o he cha wi h he lowes alue o he c i ical
Pecle numbe o lame adius (co esponding o hei noses).
Acco ding o he abo e men ioned, his means ha he in-
s abili ies g ow almos om he combus ion beginning bu
his does no co espond necessa ily wi h an ea lie appa i-
ion o a cellula lame, as explained la e .
The peninsulas ob ained o he es o uel composi ions
a e di e en depending on i hey a e plo ed e sus he Pecle
numbe o lame adius. I can be seen in Fig. 6i and 6ii ha he
pe cen age o hyd ogen a ec s signi ican ly o he s abili y
limi s bu no in a p opo ional way. Compa ing he maps
plo ed e sus he Pecle numbe and e sus he lame adius
(Fig. 6i and 6ii) i is possible o see he in luence o he lame
on hickness, especially o he combus ions wi h 80% and
100% o hyd ogen, since hey ha e bigge Pecle numbe s o
he same adii ( hinne lame on hickness). In Fig. 6ii i can
be obse ed ha he de elopmen o ins abili ies occu s a
smalle adii o he medium and high hyd ogen con en mix-
u es (50, 80 and 100% o H
2
) han o hose wi h high me hane
con en (20 and 0% o H
2
). Con as ing Fig. 6i and 6ii shows ha
he lame hickness dec eases wi h inc easing hyd ogen con-
en : o he same Pecle numbe a smalle adius is ob ained.
Ins abili y peninsulas e sus Pecle numbe o combus-
ions unde lean- uel condi ions a e p esen ed in Fig. 6iii, 6 .
Fi s hing o no ice is ha he expe imen s wi h low hyd ogen
con en a e sligh ly displaced o he le compa ed o hose
wi h s oichiome ic- uel condi ions (i.e., sligh ly smalle
alues o he c i ic Pecle numbe ). As o he medium and
highe hyd ogen con en mix u es, hei peninsulas a e
comple ely di e en since now hey a e uns able combus ions
om hei o igin. The mo phology o his cu es is jus a line,
meaning ha he g ow h a e cu es (S¼ ðnÞ) only ha e one
oo , i.e., only one alue o he wa enumbe (n) ul ils S¼0
(ins ead o he wo ha he s able combus ions ha e wi h a
peninsula shape); he uns able egion co esponds, conse-
quen ly, wi h he uppe pa o he g aph shown in Fig. 6 and
he s able egion is he in e io pa .
When he ins abili y peninsulas a e ep esen ed e sus
he lame adius (Fig. 6i and 6 i) he peninsulas o he lowe
hyd ogen con en mix u es a e displaced o he igh , i.e.,
hei c i ical adius a e bigge compa ed o hose o he
s oichiome ic case. This sugges ha mix u es wi h 0 and
20% o H
2
(i.e., me hane is dominan ) a e mo e s able a lean
condi ions han a s oichiome ic condi ions. Fo high and
medium hyd ogen con en s (dominan hyd ogen) he
endency is he same, independen ly o i he ins abili y
cu es a e ep esen ed e sus he Pecle numbe o lame
adius (see Fig. 6 and 6 i).
Flame s abili y na u e
The compa ison o he e ec i e Lewis numbe , Le
e
, (calcu-
la ed wi h mix u e p ope ies) wi h he c i ical alue Le
e *
(calcula ed by means o Eq. (7)) p o ides he key o know i a
mix u e is in insically s able o no . Fo his eason, he
alues o hese wo pa ame e s, in each combus ion p ocess,
a e plo ed oge he in Fig. 7. All he s oichiome ic mix u es
s udied in his esea ch wo k a e s able in hei o igin, since
all he Lewis numbe s ep esen ed in Fig. 7 (i) a e bigge han
i s c i ical alue (Le
e
>Le
e *
). Compa ing his Figu e wi h he
ins abili y peninsulas (Fig. 6) i can be said ha he close he
wo Lewis numbe a e, he mo e o he le he peninsula
is and he less s abilizing capabili y he he mal-di usi e e -
ec has.
The e olu ion o he Lewis numbe and i s c i ical alue
based on he hyd ogen con en can be ela ed o he dominan
uel in each case. T adi ionally, o mix u es o me hane/
hyd ogen/ai , has been accep ed ha hyd ogen is dominan
when i s con en is 50% and highe , while in case o a lowe
hyd ogen con en me hane is dominan . This can be seen in
he esea ch o bu ning eloci ies o his ypes o mix u es
ca ied ou by Reyes e al. [36].
These esul s a e compa ible wi h he expe imen al
beha io obse ed o s oichiome ic mix u es in his s udy.
I can be seen in he s abili y maps (Fig. 6), especially in
hose ep esen ed e sus he lame adius (Fig. 6ii and 6i )
o in he Lewis numbe s (Fig. 7i), ha he shape o
Fig. 7 eLe
e
and Le
e *
e sus he pe cen age o hyd ogen in
he uel mix u e.
in e na ional jou nal o hyd ogen ene gy xxx (xxxx) xxx 9
Please ci e his a icle as: Reyes M e al., S udy and cha ac e iza ion o he ins abili ies gene a ed in expanding sphe ical lames o
hyd ogen/me hane/ai mix u es, In e na ional Jou nal o Hyd ogen Ene gy, h ps://doi.o g/10.1016/j.ijhydene.2022.05.063
giCoe icien s o Eqs. (4)-6)
lWa eleng h
FFuel/ai equi alence a io: uel/ai mass a io o e
he s oichiome ic uel/ai mass a io: F¼m =ma
m =mas oich
Densi y
sExpansion a io
SG ow h a e o an ins abili y
uPa ame e associa ed wi h hyd odynamic ins abili y
UPa ame e associa ed wi h he he mal-di usi e
ins abili y
Subsc ip s
0 Re e ence condi ions
b Bu ned
c C i ical
i Ini ial
ub Unbu ned
e e ences
[1] Vu TM, Pa k J, Kim JS, Kwon OB, Yun JH, Keel SI.
Expe imen al s udy on cellula ins abili ies in hyd oca bon/
hyd ogen/ca bon monoxideeai p emixed lames. In J
Hyd ogen Ene gy 2011;36(11):6914e24.
[2] Bech old JK, Ma alon M. Hyd odynamic and di usion e ec s
on he s abili y o sphe ically expanding lames. Combus
Flame 1987;67(1):77e90.
[3] B adley D. Ins abili ies and lame speeds in la ge-scale
p emixed gaseous explosions. Philos T ans R Soc London, Se
A: Ma h Phys Eng Sci 1999;357(1764):3567e81.
[4] B adley D, Ha pe CM. The de elopmen o ins abili ies in
lamina explosion lames. Combus Flame 1994;99(3):562e72.
[5] Tinau F, Reyes M, Melga A, Gim
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