Characterization+of+Laminar+Premixed+MethanolE28093air+Flames

7/30/2019 Characterization+of+Laminar+Premixed+MethanolE28093air+Flames

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Characterization of laminar premixed methanolair flames

S.Y. Liaoa,b,*, D.M. Jianga, Z.H. Huanga, K. Zenga

a State Key Laboratory for Multiphase Flow in Power Engineering, Xian Jiaotong University, Xian 710049, Peoples Republic of Chinab Department of Power Engineering, College of Chongqing Communication, Chongqing 400035, Peoples Republic of China

Received 30 September 2005; received in revised form 15 December 2005; accepted 16 December 2005

Available online 7 February 2006

Abstract

This study focuses on the effects of initial temperature and pressure on the propagation characteristics of laminar premixed flame of methanol

air mixtures. Spherically expanding laminar premixed flames, freely propagating from spark ignition sources in initially quiescent methanolairmixtures, are continuously recorded by a high-speed CCD at various equivalence ratios and temperatures. The flames are then analyzed to deduce

the flame speed. The stretch imposed on the spherical flame front is explored experimentally; as a consequence, the unstretched laminar burning

velocities of methanolair flames have been derived. The present measurements are compared with the experimental data reported previously, and

good agreements are obtained. Combined previous results, a correlation in the form ofulZuloTu=Tu0aTPu=Pu0

bp has been developed to describe

the dependences of initial temperature and pressure on the burning velocities of methanolair flames. The global activation temperatures are

determined in terms of the burning mass flux. And then, the Zeldovich numbers for methanolair flames are estimated as a function of equivalence

ratio. On the basis of the mass burning flux, an alternative correlation of laminar burning velocities has been proposed, and agreements can still be

found in the comparison between this alternative forms and the power law correlation above.

q 2006 Elsevier Ltd. All rights reserved.

Keywords: Methanol; Laminar burning velocity; Premixed flame; Mass burning flux

1. Introduction

Methanol has been demonstrated to be one of the

promising alternative fuels for spark ignition engines. The

most important characteristic of methanol is that it is

undoubtedly the cheapest liquid alternative fuel per calorific

unit, which can be produced from the widely available

fossil raw materials including coal, natural gas and bio-

substances, that it can be produced from biological sources

and therefore represents a renewable energy source [1,2].

This essentially means that many countries can solve their

energy imbalance problems due to petroleum shortage by

using methanol as a source of energy.

It is known that, laminar burning velocities are fundamen-

tally important in regard to developing and justifying the

chemical kinetics mechanism of fuel, as well as in regard to

predicting the performance and emissions of the internal and

external combustion system [3]. Generally, chemical kinetics

of flames can be studied by performing numerical calculation

by solving balance equations of one-dimensional laminar

flames, but the chemical kinetic data in such models are not

always sufficiently well known to be used in confidence. In

practice, the measured laminar burning velocities are

commonly used to validate these chemical kinetic schemes.

There are many techniques for experimentally measuring the

laminar burning velocity of combustible mixture, such as

counterflow double flames [4], flat flame burner [5], heat flux

method [6], and closed bomb technique [714]. For spherically

expanding flames in closed combustion bomb, the stretch

imposed on the premixed flame front is well defined.

Furthermore, the asymptotic theories and experimental

measurements have suggested a linear relationship between

flame speeds and flame stretches [15]. As we know, the term of

laminar burning velocity is generally defined for one-

dimensional planar flames in theory, where flames are

unstretched. Therefore, measured flames can then be used

systematically to determine the fundamental burning velo-

citythe unstretched laminar burning velocity by mean of an

extrapolation at zero stretch [3]. Meanwhile, Markstein lengths

can, in principle, also be deduced. Here, Markstein length

characterize the variation in the local flame speed due to the

influence of external stretching, which is important in

Fuel 85 (2006) 13461353

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0016-2361/$ - see front matter q 2006 Elsevier Ltd. All rights reserved.

doi:10.1016/j.fuel.2005.12.015

* Corresponding author.

E-mail address: [email protected] (S.Y. Liao).
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expressing the onset of flame instabilities and the stretch

influence on flame quenching as well.

The measurements of burning velocities have been

extensively studied for a wide variety of hydrocarbon fuels,

including methane, ethane, propane, butane, and octane etc.

However, as to methanol, it has not been studied so far in the

literatures. Metghalchi and Keck, Gulder, and Saeed and Stonemeasured the burning velocities of methanolair mixture in

closed combustion bombs. However, in those previous studies,

a continuous observation of flame growth were not always

implemented and the laminar burning velocities were

generally, determined by resolving various combustion models

based on combustion pressure traces, but those procedures

commonly ignored the influence of stretch imposed on the

flame, hence the Markstein lengths have not been obtained as

well. Meanwhile, discrepancies still exist, to some extent,

within these experimental results. In addition, Gibbs and

Calcote [16], Muller et al. [17], and Westbrook and Dryer [18]

conducted flame chemical kinetics computations to determinethe laminar flame speed, however, the results obtained still

appear to show somewhat obvious scatterings.

In view of considerations above, the motivation of this work

is mainly due to that the burning velocity data of methanol is

still scare and less accurate. This work presents experimental

measurements of the laminar burning velocities of methanol

air premixed flames. Spherically expanding laminar premixed

flames, freely propagating from spark ignition sources in

initially quiescent methanolair mixtures, are continuously

recorded by a high-speed CCD at various equivalence ratios

and temperature conditions. The flames are analyzed to

estimate flame size, consequently, the flame speeds are derived

from the variations of the flame size against time elapsed.

Following the linear relation between flame speed and flame

stretch, the unstretched laminar burning velocities and

corresponding Markstein lengths of flames have been deduced.

The effects of the fuel/air equivalence ratio, initial temperature

and pressure on the laminar flame propagation have been

studied in detail. As a result, empirical formulas for the laminar

burning velocities are suggested, and comparison with the

previous results is made as well.

2. Experiments and procedures

Shown in Fig. 1 are the schematic diagrams of the

combustion bomb and the optical system used for recording

the flame growth. The combustion bomb has an inside size of

108!108!135 mm, as shown in Fig. 1A. Two sides of this

bomb are transparent to make the inside observable; these sides

are to provide the optical access, and the other four sides are

enclosed with resistance coils to heat the bomb to the desired

preheat temperature. The inlet/outlet valve is used to let fresh

air or combustion product in or out. The liquid fuel needed is

pre-calculated corresponding to the given equivalence ratioand then injected into the combustion chamber using a small

capability injector. The normal function of the motion of

perforated plate is to provide a turbulent combustion

environment if needed, where it is only used to enhance the

vaporization of methanol and make the reactants mixed well.

Two extended stainless steel electrodes are used to form the

spark gap at the center of this bomb. A conventional battery-

coil ignition system is used for producing the spark. The history

of the shape and size of the developing flame kernel is recorded

by a REDLAKE HG-100K high-speed CCD camera, operating

at 5000 pictures per second with a schlieren optical system, as

given in Fig. 1B. Dynamic pressure used to determine burned

gas and unburned gas properties herein, is measured since

spark ignition with a piezoelectric absolute pressure transdu-

cer, model Kistler 4075A, with a calibrating element Kistler

4618A.

Fig. 1. Schematic diagram of experimental system.

S.Y. Liao et al. / Fuel 85 (2006) 13461353 1347


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The laminar burning velocity can be deduced from the well-

established expanding flames method as described in [7]. In

this approach, the stretched flame velocity, Sn, is derived from

the flame radius versus time data t as,

SnZdrudt

(1)

where flame size ru is defined as the schlieren flame size. The

definition of the flame stretch, a, of a flame front in a quiescent

mixture is given by,

aZ1

A

dA

dt(2)

where A is the flame front area. For a spherically outward

expanding flame, the flame stretch is well defined as,

aZ1

A

dA

dtZ

2

ru

drudtZ

2

ruSn (3)

Following an early idea of Markstein, it is suggested a linear

relationship between flame speeds and flame stretches, given as

Eq. (4),

SlKS

nZL

ba (4)

Thereby, the unstretched flame speed, Sl, can be obtained as

the intercept value at aZ0, in the plot of Sn against a. The

value of burned gas Markstein length, Lb, can also be obtained

from the plot of experimental values of Sn against a, as

mentioned in Eq. (4), using linear regression shown in

Appendix A.

For the observation limited to the early stage of the flame

expansion, where the pressure does not vary significantly yet,

there is a simple relationship linking the spatial flame velocity,

Sl, to the fundamental one, i.e. the unstretched laminar burning

velocity, ul, as

ulZrbSl=ru (5)

Here, ru is the density of the unburned gas and rb that of the

burned gas, which can be determined from a quasi-dimensional

two-zone combustion model [19].

3. Results and discussions

After the spark occurs, ignition takes place in the center of

the chamber, and the flame propagates spherically to the whole

mixture. Shown in Fig. 2 is a typical case of flame propagation,

it can be seen that the schlieren flame images show a disk

whose diameter increases with time. As far as the visualization

of flame propagation was concerned, the key aim is to obtain

the flame propagating speed. The flame image analysis can

then be performed to deduce the evolution of flame radius

versus time; hence the spatial flame speed is evaluated as the

rate of variation in flame size (Eq. (1)). In order to characterize

the flame propagation, one can plot the evolution of the flame

speed in function of the flame radius, as shown in Fig. 3. As we

know, the role of spark is to initialize a flame, which is to

overcome the tendency for the flame to quench because of the

high curvature stretch rate, during the early stage of flamepropagation. It is obvious that the flame speeds illustrate a

decrease initially, and then increase gradually as the radius

increases, because of the relatively higher flame stretch and

heat loss during the early stage of the flame propagation [7].

The effects of the fuel/air equivalence ratio and preheated

temperature on the flame speed are studied in this figure. It is

clearly shown that the increase in flame speed with radius is

Fig. 2. Typical growing schlieren flame kernels for methanolair mixture with fZ1.2. The time interval is 4 ms, TuZ358 K, PuZ0.1 Mpa.

Fig. 3. Variations of Sn with flame radius, ru for methanolair mixtures

(0.1 MPa), and (A) equivalence ratio, f, (B) initial temperature, Tu.

S.Y. Liao et al. / Fuel 85 (2006) 134613531348


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much greater for the stoichiometric flames and high tempera-

ture can accelerate the flame propagation.

The local stretch imposed on the spherical flame can beevaluated via Eq. (3). Thereby, the spatial flame velocity can

be plotted as a function of the stretch rate, to characterize the

stretch effect on the flame propagation. Shown in Fig. 4 is a

selection of experimental data showing the variations of flame

speed, Sn with total stretch rate, a, at different equivalence

ratios and initial temperatures. According to Eq. (4), the

unstretched flame speed can be obtained by a zero stretch

extrapolation, also shown in Fig. 4 with solid lines, and the

slope of straight line indicates the burned gas Markstein length

Lb. Then, the fundamental burning velocity, unstretched

laminar burning velocity can be derived from the value of

the unstretched spatial flame speed via Eq. (5). Fig. 4 shows

that the flame accelerates with the decrease of stretch rate as the

flame propagates, which denotes positive burned gas Markstein

lengths for all tested flames. This figure also presents that the

flame speed of mixture with fZ0.8 is more sensitive to the

flame stretch than that of richer flames, relatively. However, it

also appear that there is no obvious difference between mixture

offZ1.0 and 1.2. This can be an indication that the curves of

mixture with fZ1.0 and 1.2 are relatively flatter, compared to

that of flames of fZ0.8 (Fig. 4A). The effect of initial

temperature on the flame propagation is investigated as well.

The increasing temperature results in a corresponding increase

in the flame speed. Shown in Fig. 4B is the variation of flame

speed with stretch for stoichiometric flames at different initial

temperature conditions. However, apart from the overall

magnitude of the flame speed, there appears little influence

of temperature on flame behaviour because it is indicated thatthe slopes of flame speed curves do not change obviously.

Fig. 5 exhibits the influences of fuel/air equivalence ratio

and initial temperature on the flame/stretch interaction,

namely, quantified by the burned gas Markstein lengths Lb.

With regard to the values of Lb of methanolair flames, few

experimental results have been reported previously. Our

measurements report that the value of Lb decreases slightly

with the increasing equivalence ratio, however, when fO1.0,

the value ofLb almost appears no change with the variation of

the equivalence ratio. The effect of the preheat temperature on

Lb can be neglected, as it is shown that Lb varies little for

different temperature conditions. Generally, all of the

Markstein lengths are positive. This can be demonstrated bythe fact that the measured flames are smooth, in which no

instability occurs during the atmospheric pressure experiments,

for the present range of fuel/air equivalence ratio.

Shown in Fig. 6 are the measured laminar burning velocities

for methanolair flame at 358 K and the atmospheric pressure,

over a wide equivalence ratio range. In this figure are also

plotted the results obtained previously for comparison.

Obviously, a good agreement of our measurements with

f=0.8

f=1.0

f=1.2

a/s1

Fig. 4. Variations of flame speeds with different stretch rates, and (A)

equivalence ratio, f, (B) initial temperature, Tu.

Fig. 5. Measured Markstein lengths of methanolair flames as a function of

fuelair equivalence ratio. The curves are second order fitting to measured

points.

Fig. 6. Experimental laminar burning velocities for methanolair flames at

358 K and 0.1 MPa.

S.Y. Liao et al. / Fuel 85 (2006) 13461353 1349


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literature values of Metghalchi and Keck can be found, while

we slightly underestimate the laminar flame velocity for rich

mixtures, compared to those obtained by Saeed and Stone.

According to these results, at normal pressure of 0.1 MPa and

temperature of 358 K, the maximum burning velocity for

methanolair flame appears to be approximately between 54

and 61 cm/s for Metghalchi et al. and Saeed et al., respectively,whereas the present measured value is about 56.2 cm/s. Over

the measured range of the equivalence ratio, the present results

can be fit by a second order polynomial of the equivalence ratio

as Eq. (6), also shown in Fig. 6 with a solid curve.

ul0ZK195:6f2C419:92fK169:43 (6)

where the subscript o represents the reference conditions, i.e.

358 K and 0.1 MPa in this work. The maximum burning

velocity in Eq. (6) is about 55.9 cm/s at an equivalence ratio of

1.07, corresponding to 56.2 cm/s of experiment. Generally

speaking, one can see that, the fitting burning velocities showgood agreements with the results of experiment over a wide

range of equivalence ratio. In particular, the study of the effect

of temperature on the burning velocities has been emphasized

in this work. Fig. 7 illustrates temperature dependence of the

laminar burning velocities of methanol at atmospherical

pressure. More often, this dependence generally is expressed

as a simple power law relation of nondimensional temperature

(Tu/Tu0) and nondimensional pressure (Pu/Pu0), at the datum

conditions, as

ulZuloTu=Tu0aTPu=Puo

bP (7)

where aT and bP represent the parameters of temperature and

pressure dependences. The power law relation is also presented

in Fig. 7 as solid curves, where initial pressure of premixed

mixture has no change. It can be seen that the plotted curves of

the laminar burning velocities exhibit good agreements with

experimental data over the equivalence ratio range. The

parameter aT is clearly influenced by the fuel/air equivalence

ratio. Table 1 lists the present results for aT. Moreover, in this

table are also presented the results reported previously for

comparison. Metghalchi and Keck reported that aT is function

of the equivalence ratio, which was formulated as aTZ2.18K

0.8 (fK1) at 300 K and 0.1 MPa, while Gulder was given a

constant of 1.75. Although slight differences can be found in

our results against previously studies, partially because of thedifference of the datum temperature, a similar variation of

parameter aT against equivalence ratio can also be obtained, as

given in Eq. (8),

aTZ1:85K0:6fK1 (8)

Fig. 7 also shows the comparisons of the burning

velocities for methanolair flames at various temperatures

derived from Eq. (7) with those derived from the published

empirical formulas. On the whole, it is found that our

predictions are more consistent with those of Metghalchi and

Keck, and especially, better agreements can been obtained

for the mixture at relatively low temperature. However, theresults of Saeed and Stone gradually underestimated the

laminar burning velocities with the increase of initial

temperature.

The pressure effect on the laminar burning velocities has not

been investigated experimentally in this work. Metghalchi and

Fig. 7. Laminar burning velocity of methanolair flames on temperatures at

different equivalence ratios.

Table 1

Evolution ofaT with equivalence ratio

aT(Metghalchi and Keck)

aT(Gulder)

aT(present study)

fZ0.8 2.47 1.75 1.98

fZ1.0 2.11 1.83

fZ1.2 1.98 1.75

S.Y. Liao et al. / Fuel 85 (2006) 134613531350


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Keck, Saeed and Stone had reported that the pressure

dependence, bP is weakly influenced by the fuel/air equival-

ence ratio. On the basis of the available data reportedpreviously (Fig. 8), we can obtain an empirical relation to

describe the dependence bP on the fuel/air equivalence ratio, as

given in Eq. (9).

bTZK0:1651C0:2fK1 (9)

Thereby, combining Eqs. (6), (8) and (9), Eq. (7)

provides an approach to empirically estimate the laminar

burning velocities of methanolair mixtures by functions of

temperature, pressure and equivalence ratio. In order to

validate these empirical formulas, it is informative to

compare its behaviour over more wide temperature and

pressure ranges, with those reported previously, as shown inFigs. 9 and 10. It is known that, high temperature can

accelerate the rate of combustion chemistry, thus, the

laminar burning velocities increase consistently with the

increase of initial temperature of combustible mixtures. In

Fig. 9, we can find that, the predictions of Metghalchi and

Keck are more sensitive to temperature changes. When

temperature varies from 358 to 480 K, the derivations

among these empirical relations relatively become signifi-

cant, showing that the predicted results of Metghalchi and

Keck are slightly overestimates the laminar burning

velocities than the others at 480 K. The effect of initial

pressure on the laminar burning velocities is presented in

Fig. 10. The general trend is that, the increase of pressure

results in the decrease of the laminar burning velocities. No

direct experimental comparison has been made herein.Similarly, the available predicted results of Metghalchi

and Keck, Saeed and Stone are presented for comparison.

The overall agreement among these results is very good

over the whole range of equivalence ratios.

It is suggested that the expression of the laminar burning

velocity in terms of the square root of an Arrhenius expression

in adiabatic flame temperature Tb with overall activation

energy E, has more significant theoretical basis than does the

power law correlations as given above [2022]. Peters and

Williams [21] has derived an asymptotic structure of the flame

that introduces the inner layer temperature T0 in the fuel

consumption, and express the activation temperature E/R in

terms of the mass flux (ruul) and adiabatic flame temperature Tbas following,

E

RZ2T2b

dlnruul

dTb(10)

where R denotes gas constant and an assumption that T0

remains constant was adopted. In practice, Eq. (10) can also be

rewritten as a alternative form as

E

RZK

d2lnruul

d1=Tb(11)

This indicates that the activation temperature E/R can be

derived directly from the linear plot of 2ln(ruul) against 1/Tb.

Meanwhile, Eq. (11) also suggests that the evaluation of E/R

requires the determination ofruul in advance. It is known that,

for a given mixture with Tu, the adiabatic flame Tb and the mass

burning flux ruul can be determined from chemical equilibrium

calculation and Eqs. (6)(9). That is to say, the power law

relations of Eqs. (6)(9) just provide an approach to determine

activation temperature of flames.

Plots of 2ln(ruul) against 1/Tb are shown in Fig. 11. The

activation temperature at each pressure is given by the slope of

appropriate straight fitting line. It is found that E/R of

methanolair flames is a weak function of temperature, while

strongly depends on initial pressure of mixtures, which appears

Fig. 8. Parameter bP for methanolair mixtures, the straight line shows the least

squares fit for Eq. (9) to points.

Fig. 9. Effect of temperature on burning velocities for methanolair flames.

Fig. 10. Effect of pressure on burning velocities for methanolair flames.

S.Y. Liao et al. / Fuel 85 (2006) 13461353 1351


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to have a similar behaviour for methaneair [7] and iso-octane

air flames [1]. Over the studied range, E/R can be

approximately expressed by,

E

RZ2566PuC16085 (12)

With the pressure Pu in MPa, and the intercept value of

fitting lines at 1/TbZ0, C is found to be,

CZ10:88P0:253u (13)

Integration of Eq. (11) gives:

lnruulZKE

2R

1

TbC0:5C (14)

Or rearrangement gives:

ulZ exp KE

2R

1

Tb

exp0:5C

ru(15)

When E/R and Cpresumed from Eqs. (12) and (13), Eq. (15)

gives an alternative correlation of laminar burning velocities to

that presented by Eq. (7). The calculated laminar burning

velocities using Eq. (15) together with Eqs. (12) and (13) are

also plotted in Figs. 9 and 10. We can see that, the predicted

burning velocities of Eq. (15) exhibit good agreements withthose of Eq. (7), and still has no significant deviation with the

predicted results by Metghalchi and Keck, and Saeed and

Stone. Certainly, Eq. (15) is directly derived from the empirical

correlation of Eq. (7), rather than experimental measurements,

that is to say, the former can only be thought as an alternative

form of the later, the good agreement for Eq. (15) just

demonstrates that the derivation in deduction of Eq. (15) from

Eq. (7) is acceptable. However, as mentioned above, the former

has more physical relevance, which just is the significance of

this procedure.

From the values of E/R, the Zeldovich number, b can be

determined [7] as,

bZE

2RT2bTbKTu (16)

Here b represents the sensitivity of chemical reactions to the

variation of the maximum flame temperature and the inverse of

it denotes an effective dimensionless width of the reaction zone

[20]. Shown in Fig. 12 are the variations ofb against pressure

for methanolair mixtures. The values ofb is clearly influenced

by equivalence ratio, temperature and pressure. The increase of

pressure results in increasing b, because of the increase of inner

flame temperature T0 [17]. Also, it is out of question that, b

attains a minimum for increased Tu and near-stoichiometric

mixture, which has a maximum Tb. More often, another form of

Zeldovich number, Ze, defined as

ZeZbTbKTu

T0KTu(17)

The predicted results of Ze for methanolair are plotted in

Fig. 13, and the values of methaneair and iso-octaneair

mixtures are presented for comparison. The present results and

those of methaneair generally show weak dependence on

pressure than those computed values for iso-octaneair

mixtures. And methanolair gives a similar tendency against

the variation of equivalence ratio with methaneair flames,

which is obviously contrast to iso-octane mixture in air.

Fig. 12. Variations of b with the pressure for methanolair mixture with

different equivalence ratio.

Fig. 11. Plots of 2ln(ruul) against 1/Tb for methanolair mixtures. Where full

lines are linear fits through the points obtained from Eqs. (6)(9), and dashed

lines denote burning mass flux correlation of Eq. (14).Fig. 13. Variations of Ze with the pressure for methanolair mixture with

different equivalence ratio, and the results of methaneair [7] and iso-octane

air [17] are presented for comparison, where &, 0.1 MPa; B, 0.5MPa; 6,

1.0 MPa.

S.Y. Liao et al. / Fuel 85 (2006) 134613531352


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4. Conclusions

In this paper, centrally ignited, spherically expanding

flames have been measured using schlieren photography to

determine the influence of fuel/air equivalence ratio and

initial temperature on the laminar burning velocities of

methanolair mixtures. The stretch imposed at the flamefront has been explored experimentally and the Markstein

lengths are estimated to characterize its effect. Combined

the measurements and previous results, an empirical relation

by the function form ulZuloTu=Tu0aTPu=Pu0

bT , has been

obtained, to formulate the laminar burning velocity

dependencies on the equivalence ratio, initial temperature

and pressure. On the basis of the asymptotic theory of flame

structure, an alternative burning velocities correlation has

been derived from above power law formula, and the

expressions for mass burning flux, Zeldovich numbers have

been obtained. The comparisons of our empirical corre-

lations for burning velocities with the results reported

previously show a good agreement, which validates the

present study.

Acknowledgements

This work is supported by the state key project of

fundamental research plan of Peoples Republic of China

(No. 2001CB209206), and the key project of NSFC (No.

50156040).

Appendix A. Linear regression

Consider Npairs of computed or measured values of (a, Sn)i.

They satisfy Eq. (4):

SlKSnZLba (A1)

The value ofLb is calculated from

LbZK

PNiZ1

aiK aSniKSn

PNiZ1

aiK a2

(A2)

in which

Sn Z1

N

XNiZ1

Sni; aZ1

N

XNiZ1

ai; SlZ SnCLb a (A3)

The standard derivation is given by

eZ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

NK2

XNiZ1

SnKSn 2CLb

XNiZ1

aiK aSniKSn

" #vuut(A4)

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