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Thermophysics and Aeromechanics, 2013, Vol. 20, No. 6

Heat exchange in boundary layer on permeable plate at injection and combustion*

V.V. Lukashov1,

2, V.V. Terekhov1, and K. Hanjalič 2,

3

1Kutateladze Institute of Thermophysics SB RAS, Novosibirsk, Russia 2Novosibirsk State University, Novosibirsk, Russia 3Delft University of Technology, Delft, The Netherlands

E-mail: [email protected]

(Received April 29, 2013)

The peculiarities of the heat and mass exchange in a laminar boundary layer with combustion at the injection of the fuel mixture H2/N2 through the permeable surface are considered. It is shown that at a certain value of the injection parameter, the value of the heat flux into the wall averaged over the length has a maximum. An analytic estimate is proposed for determining the maximum heat flux at the combustion depending on the injection intensity. The obtained relations agree with the results of experimental studies and numerical modelling.

Key words: boundary layer, injection, combustion, heat-flux maximum.

Introduction

Investigations of heat and mass exchange in reacting flows with gas injection through the porous bounding walls have often served as a convenient substitute for studying evaporation (pyrolysis) and subsequent combustion in a boundary layer over liquid and solid fuel surfaces. The problem is also relevant to a wide spectrum of other applications, such as thermal protection of gas-turbine blades and walls of combustion chambers and other high-temperature devices. The actuality and relevance of the prob-lem is confirmed by a large number of publications in the literature, some of which are worth discussing.

Emmons [1] was among the first to obtain a solution for the boundary layer with combustion on the evaporating surface by using a self-similar approximation of the boun-dary layer (with the Schwab—Zeldovich model of the flame front). In particular, it was shown that the heat flux into the wall decreases proportionally to the square root of the distance from the leading edge. The Emmons model was extended by Kikkava and Yoshikava [2] for the case of the propane uniform injection into the boundary layer with regard for real proper-ties of the gaseous mixture (the Lewis number Le ≠ 1). It was shown, apparently for the first time, that the consideration of the variable properties due to the variation of the mixture composi-tion may lead to a front temperature reduction down to 700º and to the variation of the front

* The work was partially supported by the Ministry of Education of Russian Federation (Contract 14.518.11.7015) and of the Russian Federation government (Grant No. 11.G34.31.0046).

© V.V. Lukashov, V.V. Terekhov, and K. Hanjalič, 2013

V.V. Lukashov, V.V. Terekhov, and K. Hanjalič

688

vertical coordinate down to 20 %. The domain of the Emmons model applicability was studied by Raghavan et al. [3], where it was, in particular, shown by the example of the methanol evaporation and combustion that in the region of a stable combustion at the velocities of the air flow around the permeable plate above 0.5 m/s, the fuel evaporation intensity is related linearly to the skin-friction coefficient on the wall.

The use of the self-similar description of the boundary layer has its limitations. In particular, it does not work near the flame leading edge, where there is a singularity at such an approach (the heat flux tends to infinity). The n-pentane combustion was studied by Ananth et al. [4] at a porous injection into the laminar boundary layer. The unsteady Navier—Stokes equations were used. The heat release intensity was shown to be maximal near the flame leading edge. Its magnitude then decreased rapidly downstream.

Despite the achieved progress, a number of questions remain unsolved. One of them, which was posed by the Academician E.P. Volchkov, is the determination of the condi-tions for existence of the maximum or minimum of the heat flux into the wall at the fuel injection through a porous plate into the boundary layer at its combustion. It is intuitively clear that depending on the fuel flow rate there must exist a regime, in which the heat flux into the wall has its extremum. In fact, there is no combustion at a zero fuel flow rate and, as a consequence, there is no heat release. In the case of a fairly high injection intensity, the combustion front and the boundary layer are pushed from the wall and upon reaching a critical injection, the boundary layer is pushed away from the wall, which again leads to a zero heat flux into the wall. It is obvious that there must be a point of the maximum heat flux into the wall between the zero and critical flow rate of the injected gas. The problem of the location and of the extremal heat flux at the injection into the non-reacting boundary layer was solved by Volchkov et al. [5]. It was shown that the temperature and concentration gra-dients on the wall with increasing injection at first grow, reaching their maximum, then drop, whereas the velocity gradient monotonously decreases in the entire range of the injection intensity. The location of the heat flux maximum depends on the relation of the molecular weights of the injected gas and main flow.

The purpose of the present work is the extension of the proposed approach for the case of a flow with combustion. In experiments with the injection and combustion of fuel mix-ture H2/N2, we consider the peculiarities of the heat and mass exchange in a laminar boundary layer. The influence of injection on the magnitude of the heat flux on the wall is studied analytically and numerically.

Experimental setup and investigation technique

The experimental investigations were conducted in a subsonic wind tunnel (Fig. 1) with the 105×105 mm section of the channel at the test section inlet. The velocity of air stream over the plate was varied in the range from 1 to 10 m/s. The fuel considered was a mixture of hydrogen and nitrogen, which was injected uniformly into the boundary layer through a horizontal porous plate of planform size S = 95×145 mm. The hydrogen mass

Fig. 1. Experimental setup diagram. 1 ⎯ porous plate, 2 ⎯ flame stabilizer, 3 — measuring probe.

Thermophysics and Aeromechanics. 2013. Vol. 20. No. 6

689

fraction 2

1HK in the fuel was specified from 2 to 10 %. The regulators of the gas flow rate

manufactured by MKS Instruments were used for preparing the fuel mixture. The error in de-termining the fuel mixture composition did not exceed 0.4 %± .

The porous plate 1 was made of the ceramic material TZMK-10 based on amorphous quartz fiber or its analogue Li-900 produced in the USA. It has a low coefficient of ther-mal conductivity ⎯ 0.05 W/m/K and possesses a high porosity of 90−95 %.

The wall material is capable to withstand great many exposures to the temperatures up to 1370 K. while preserving the dielectric properties. This made it possible to place chromel-alumel thermocouples of 100 μm in diameter in the porous body and thus to meas-ure the surface temperature without using electric insulation. In our experiments, the sur-face temperature values were below 1100 K.

The numerical modeling was conducted in parallel with experimental research. This enabled to extend the range of studied parameters as well as to obtain a number of impor-tant quantities, which either could not be measured or their experimental evaluation is asso-ciated with difficulties and high measuring uncertainly (e.g. the concentrations of gaseous mixture components on the wall).

The mathematical model employed in the present work is based on the unsteady two-dimensional equations of Navier—Stokes, energy and diffusion as well as chemical kinetics. Diffusion fluxes were determined by the Stefan—Maxwell model with regard for thermal dif-fusion effect. The use of such an approach to the determination of fluxes at a multicomponent diffusion is very important for modeling of laminar hydrogen flames. The kinetics of hydrogen combustion in air was determined according to the GRI-Mech 3.0 model without considering the reactions involving the nitrogen. Because of a fairly low level of temperatures the radiation was taken into account only in the form of heat losses from the wall.

The numerical realization of the mathematical model was based on the control volume method on a structured non-staggered grid. The convective terms were discretized by a second-order upwind scheme and the diffusion terms by the central-differencing scheme. The coupling of the velocity and pressure was handled by the SIMPLEC algorithm, and the obtained systems of algebraic equations were solved by a strongly implicit procedure.

The boundary conditions at the inlet section were specified from the profiles measured in the experiment. Outflow conditions are specified at the upper and right boundaries on which all variables were computed by extrapolation, the tangential velocity on the wall equaled zero, the normal component was assumed given. The values of the temperature and concentrations of species were computed from the balance relations. The ignition was ensured by specifying a high wall temperature at the initial stage. The grid-independent solution was reached

with the 250×200 nodes distributed non-uniformly, clustered in the regions of high gradients. A comparison of modeling results with experimental data obtained in the present work as well as with the ones presented earlier by other authors was presented in Volchkov et al. [6].

Determination of the conditions of the maximum heat flux from balance relations

Without considering the radiation one can define the balance of heat fluxes on a per-meable surface in the form

w w 1 w 1w

( ),pT

q J C T Ty

⎛ ⎞∂− = = −⎜ ⎟∂⎝ ⎠λ (1)

where Tw is the temperature of the wall surface, Т1 is the temperature of the injected gaseous

mixture, Ср1 is the heat capacity of injected gaseous mixture.


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The temperature difference w 1T T− is conveniently expressed as

ww 1

w 1

.p

qT T

J C− = (2)

Using the definition of the thermal Stanton number StT, one can present the difference 0 wT T− as

w0 w

0 0 0

,Stp T

qT T

U C− =

ρ (3)

where Т0 is the main flow temperature, ρ0 is the main flow density, U0 is the velocity

in the flow core, Ср0 is the main flow heat capacity. Adding (2) and (3) we obtain:

0 1 ww 1 0 0 0

1 1

Stp p T

T T qJ C U C

⎛ ⎞− = +⎜ ⎟

⎝ ⎠ρ or ( )

w

0 0 00 0 0 0 1

w 1

1.

1

St

pp

p T

qU CU C T T

J C

=− +

ρρ (4)

We denote the left-hand side of equation (4) by St .K This is the relative heat flux, where

the difference 0 1T T− stands in the denominator in contrast to the Stanton number. One can

transform expression (4) for convenience of the further analysis using the relative function of heat exchange ΨТ:

00

1

St 1.

1 1StT

Tp

pT T

qC

b C

= =+

Ψ

(5)

These relations are valid both for combustion and chemically inert systems. As shown by expe-riments of Volchkov and Lukashov [7], in the case of combustion it is more appropriate and convenient to use as the reference temperature in the Stanton number the temperature of the combustion front. Admittedly, such an approach to heat exchange description in combus-tion has some peculiarities and restrictions, which were analyzed in detail by Volchkov et al. [8]. It is noted that the relative function of heat exchange in expression (5) ( ) **0 Re

St / StT

T TΨ = has

been taken for the condition **ReT = idem, here St0 is the Stanton number describing the heat

exchange for the conditions on the impermeable wall. Expression (4) can be transformed also with the aid of ( )0 Re

St / St .x

x T xΨ = Then for the laminar boundary layer, using the relation

from Isaev et al. [9] between xΨ and ,TΨ we have

1.x T

T T

q qb

=Ψ +

(6)

Relation (5) holds both for the laminar and turbulent regimes of the boundary layer flow. However, one should bear in mind that it is necessary to use the appropriate form of the relative function of heat exchange TΨ that corresponds to the flow regime under consi-

deration. For the laminar boundary layer with injection one can use the linear approximation proposed by S.S. Kutateladze:

cr1 ,T T Tb bΨ = − (7)

where ( )2 3cr1 23.4 / ,Tb μ μ= μ1 and μ2 are the molecular masses of the gaseous mixture com-

ponents. Note that in the case of boundary conditions w const( )J x= , the crTb value presented

in the work [10] increases by factor 2 . Figure 2 shows the general form of the dependence q on Tb computed from (5) and (7).

As seen from the presented data, the location of the maximum heat flux depends on the relation


691

of molecular weights. With injection of a lighter gas, the maximum heat flux location shifts towards lower values of the injection parameter. In contrast, when injecting a heavier gas, the maximum heat flux is reached at a more intense injection.

From the condition of the existence of the maximum of

Tbq one can obtain from

relations (5) and (7) the injection parameter value at which Tq has the maximum:

cr 1 0* 0

01cr

1

1.

11

p ppT

pp

p T

b C CCb

CCC b

−=

− (8)

Using the relation of the relative heat flux to the quantity Tb (5), one can estimate the value

of the maximum heat flux on a permeable wall with the aid of relation

0 0 0 0 1w 0

0* *

1

( )St .

1 1

( )

p

p

pT T

U C T Tq

C

Cb b

−=

+Ψ

ρ (9)

Results and discussion

In the present work, the heat exchange on the wall was studied under the variation of both the magnitude of the fuel flow rate and the fuel thermophysical properties. The latter were va-ried by diluting the fuel (hydrogen) with a non-reacting gas (nitrogen). The range of the injec-

tion intensity was wJ = 0.1–3 %. The lower boundary of the injection intensity was deter-

mined by the need to maintain a stable combustion, whereas the upper boundary was limited by approaching the critical injection with a subsequent detachment of the boundary layer from the wall. The hydrogen concentration in the fuel mixture was varied from 2 % to 10 % in experiment and from 2 % to 100 % in the modelling. Note that lower fuel concen-trations lead to an unstable burning in the boundary layer, and at the concentrations below 0.6 %, the combustion is impossible not only in the diffusion regime in the boundary layer but also in a premixed mixture. It is also noted that the experiments were done for two regimes corresponding to the main airstream velocity of 2 m/s and 4 m/s. In both cases, for the plate length of 120 mm, the flow over the entire plate length remained laminar.

Unlike the injection of non-reacting substances, in stable combustion of a gaseous fuel, the heat flux into the wall will always vary non-monotonically over the length. The maximum heat flux into the wall occures near the leading edge of the permeable plate, where the flame approaches closest to the surface. Note that in this region, the gradient of parameters in the transverse and streamwise directions may prove comparable, and the boundary layer ap-proximation is inapplicable. When approaching the flame-off conditions (which is reached either by reducing the injection intensity or at the expense of reducing the combustible gas fraction in the fuel mixture with inert dilutant) the location of the heat flux maximum shifts

Fig. 2. Dependence of relative heat flux on the injection parameter.


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downstream. Figure 3 presents the dis-tributions of the wall heat flux over the plate length when combusting a mixture with 2 % of the hydrogen mass diluted by nitrogen, which con-firm the above assertions. At the relative

injection value wJ = 0.6 %, the combus-

tion was impossible, which corresponds to the flame-off limit reported by Volchkov et al. [6]

for the hydrogen-nitrogen mixtures. At wJ = 3 %, the boundary layer detached over

the most part of the surface. The wall temperature approached the room temperature, and heat fluxes became small.

Figure 4 shows the distribution of the heat flux into the wall over the plate length. It is seen that over the initial portion of the porous plate, the heat flux equals zero. This is because the diffusion boundary layer only begins to develop, and the fuel concentration is insufficient for the combustion to start-up. A sharp increase in the heat flux is observed soon downstream, corresponding to the combustion onset. At the minimum possible values of the injection inten-sity that ensure stable combustion, the temperatures in the flame front are low and, as the nu-merical modeling results show, are close to the values corresponding to the combustion near the lower concentration limit of the premixed mixture of the original fuel in air. Under these conditions, the heat flux is obviously small. As the injection intensity increases, the heat flux into the wall grows and reaches its maximum. A reduction of the heat flux is observed further downstream. This happens because when the boundary layer develops the flame front moves away from the wall, and in some regimes, the boundary layer detaches from the wall and, consequently, leads to zero wall heat flux. At a highly intense injection, the boundary layer separation from the wall starts in the trailing part of the porous plate, and the heat and mass exchange stops.

The effect of the injection intensity can be summarized as follows: the lower injec-tions correspond to lower hydrogen concentrations on the wall and, as a consequence, to a lower heat release inside the boundary layer. This leads to a reduction of the absolute value of the heat flux maximum. Besides, the flame front displacement from the wall at lower fuel flow rates is slower, as can be observed by the location of the maximum over the plate length, which shifts down-stream. Some discrepancy is noted be-tween the experimental data and numer-ical modeling at the plate start. This is attributed to the fact that the mathemati-cal model presumes no heat conduction

Fig. 3. Local values of the heat flux on the wall.

2

1HK = 2 %, U0 = 2 m/s, wJ = 0.7 % (1), 1.4 % (2),

3 % (3).

Fig. 4. Distribution of the heat flux into the wall over the plate length for different injection intensities. Symbols ⎯ experiment, lines ⎯ numerical modelling.

U0 = 4 m/s, 2

1HK = 10 %, wJ = 0.1 % (1), 0.2 % (2),

0.4 % (3).


693

inside the plate, whereas in the experi-ment, the heat flux inside the porous material is undoubtedly present and plays a certain role, especially in the region of considerable temperature gradients, which are present in the flame leading edge.

The dilution of hydrogen with nitro-gen is in many respects similar to the injection intensity reduction (Fig. 5). However, in the case of fuel mixture dilution, the location of the local maximum of the heat flux changes significantly. Note that such a comparison is not quite correct. One more parameter appears at the combustion ⎯ the minimum intensity of injection, which ensures a stable combustion of the fuel mixture of a given composition.

The maximum is always present in the distribution of the local heat flux over the length, which is the specifics of the combustion process. It is of interest to analyze the heat fluxes av-eraged over the porous plate length. In this case there is a possibility to determine analytically the influence of the injection intensity on the surface heating by using relations (1)−(9). The alteration of the averaged heat load of the surface depending on the permeability parameter

0w L/ StTb J= is presented in Fig. 6. Here we use 0

LSt 0.94 / Rex= as the standard heat ex-

change law on the wall [10]. The same figure shows the results obtained by formula (8). A similar picture is observed, as seen in Fig. 6, at different freestream velocities practically in the entire range of the fuel dilutions with non-combustible gas. The values of parameters used for computing the estimate * ,Tb at which the mean heat flux into the wall is maximum, are

presented in the table. It should be noted that near the flame-off conditions estimation (8) works less accurate.

Fig. 5. Heat flux distribution at differ-ent degrees of the hydrogen dilution by the nitrogen (experiment).

U0 = 2 m/s, Jw = 0.8 %, 2

1HK = 2 % (1), 10 % (2).

T a bl e Parameters for different fuel compositions

2

1HK = 10 %

2

1HK = 5 %

2

1HK = 2 %

1,pC kJ/kg/K 2.36 1.86 1.27

1,μ kg/kmole 12.2 17 22.2

crIb 1.91 2.38 2.85

crIIb 2.7 3.36 4.03

*Tb 0.8 1 1.2

Fig. 6. Mean heat flux on the porous surface vs. the permeability parameter value bT.

Solid curves ⎯ numerical modelling, points ⎯ experimental data, dashed line ⎯ formula (8); U0 = 4 m/s (1), 2 m/s (2).


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Conclusions

The experimental data and results of the numerical modeling of a laminar boundary layer near a porous plate with combustion of a hydrogen-nitrogen mixture in air have been pre-sented. Special attention was paid to the distribution of heat fluxes on the wall depending on the intensity of the fuel mixture injection and the degree of fuel dilution. The local heat flux in the case of combustion as well as for a uniform non-isothermal injection has been shown to reach its maximum value at a certain location on the plate. The averaged value of the heat flux into the wall also has a maximum, its value depending on the injection intensity. The technique for analyzing the maximum heat fluxes for the case of an inert non-isothermal injection, which was developed previously by E.P. Volchkov, has been modified and extended for the case of a boundary layer with combustion.

Nomenclature

W 0/ StTb J= ⎯ permeability parameter.

Cp ⎯ heat capacity of gaseous mixture, J/kg/K,

( )W 1 WJ = ρ υ ⎯ injection intensity, kg/s/m

2,

W W 0 0/J J U= ρ ⎯ relative injection intensity,

T ⎯ temperature, K, q

W ⎯ local heat flux (relation (1)), kW/m

2,

qL ⎯ mean heat flux over the length, kW/m

2,

St ⎯ Stanton criterion, Le = ρCpD/λ ⎯ Lewis number, λ ⎯ thermal conductivity coefficient of gaseous mixture, W/m/K, μ1 ⎯ molecular weight of fuel mixture, kg/kmole,

( ) **0 ReSt / St

TT TΨ = ⎯ relative function of heat exchange.

Subscripts w ⎯ on the wall, 0 ⎯ parameters in the flow core, 1 ⎯ fuel mixture, L ⎯ averaging over the length,

cr ⎯ parameters of critical injection, I ⎯ boundary conditions of the Ist kind, II ⎯ boundary conditions of the IInd kind.

References

1. H. Emmons, The film combustion of liquid fuel, Z. Angew. Math. Mech., 1956, Vol. 36, P. 60−71.

2. S. Kikkava and K. Yoshikava, Theoretical investigation on laminar boundary layer with combustion on a flat plate, Int. J. Heat Mass Transfer, 1973, Vol. 16, P. 1215−1229.

3. V. Raghavan, A.S. Rangwala, and J.L. Torero, Laminar flame propagation on a horizontal fuel surface: Verifi-cation of classical Emmons solution, Combustion Theory and Modeling, 2009, Vol. 13, No. 1, P. 121−141.

4. R. Ananth, P.A. Tatem, and C.C. Ndubizu, A numerical model for the development of a boundary layer diffu-sion flame over a porous plate, Naval Research Laboratory Memorandum Report NRL/MR/6183-01-8547, 2001.

5. E.P. Volchkov, V.V. Lukashov, and D.S. Dunaev, On the influence of boundary conditions on heat and mass exchange in boundary layer, Trudy RNKT-III, 2002, Vol. 2, P. 103−105.

6. E.P. Volchkov, V.V. Lukashov, V.V. Terekhov, and K. Hanjalič, Characterization of the flame blow-off condi-tions in a laminar boundary layer with hydrogen injection, Combustion and Flame, 2013, Vol. 160, No. 19, P. 1999−2008.

7. E.P. Volchkov and V.V. Lukashov, Experimental study of a laminar boundary layer with hydrogen combustion, Combust., Expl., Shock Waves, 2012, Vol. 48, No. 4, P. 375−381.

8. E.P. Volchkov, V.I. Terekhov, and V.V. Terekhov, Flow structure and heat and mass transfer in boundary layers with injection of chemically reacting substances (Review), Combust., Expl., Shock Waves, 2004,Vol. 40, No. 1, P. 1−16.

9. S.I. Isaev, I.A. Kozhinov, V.I. Kofanov et al., Theory of Heat and Mass Exchange: Textbook for Higher Schools, A.I. Leontiev (Ed.), Vysshaya shkola, Moscow, 1979.

10. V.M. Eroshenko and L.I. Zaichik, Hydrodynamics and Heat and Mass Exchange on Permeable Surfaces, Nauka, Moscow, 1984.

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