_{1}

The modified empirical two-temperature model of surface burning on a foam metal matrix was proposed. The comparative experimental studies of radiation properties of both matrices without and with ceramic coating (alumina) were carried out. Measurement was conducted in different spectral ranges. The experimental results were compared with theoretical calculations. It was shown that the integral radiation efficiency of the matrix with ceramic coating was comparable with radiation efficiency of the matrix without any coating in the wide range of the firing rate and surpassed it on 30% - 40% at firing rate above 50 W/cm
^{2}.

Surface burning of a gas mixture on a permeable matrix is accompanied by strong IR radiation from the matrix surface. Radiation burners on the base of the surface burning are widely used in the industry. Radiation efficiency of surface burning is efficiency of the contribution of the radiation flux from the matrix surface in the total energy balance. Its increasing is the important problem for IR-burners. It can be provided by growing the matrix surface temperature. It was shown [

However, to predict the radiation efficiency of the surface burning, i.e. contribution of the radiation flux from the surface of a foam metal matrix and its changing for matrix coated with the ceramic film are rather difficult. The difficulties are connected with founding exact physical characteristics of the matrix and properties of the ceramic coating. An alumina and zircon films have windows of optical transparency in the infrared spectral range. Therefore, integral radiation emissivity depends on the temperature, ceramics film thickness and spectral emissivity of the substrate i.e. from the matrix material. In our case, we focused on comparison of the surface burning on identic matrixes with ceramic coating and without one.

There are many theoretical publications concerning surface combustion on the uniform matrix (for example [

The simple empirical model was suggested in [_{sH}. Taking into account preheating of the gas mixture in matrix pores the empirical expression for flame rate was written as follows:

The temperature exponent n was equal of 2 for methane [

The parameter B determines also the critical value of firing rate when the burning failure occurs (impossibility of the solution (2)).

Appearance of surface_{ }permeability of the matrix η_{s} in the parameter B was stipulated by the assumption that the flame front rate in equilibrium was equal to the speed of expanded gas streams flowing out from channels or pores of the

matrix, i.e._{g}-relative cross section of_{ }gas streams. If

flame front is located close with the matrix surface at low speed of gas flow that it is possible to put η_{g} = η_{s}. If flame front is located far from the matrix surface at high speed of gas flow that the inequality η_{g} > η_{s} is possible. A correction coefficient can be demanded because of this effect. This coefficient should be determined from comparison of calculations and experiments.

Distribution of the matrix temperature T_{s} on coordinate x in the matrix body at known value T_{sH} was calculated separately by solution of the governing heat transfer equation within the framework of one-temperature model [

Coordinates x = 0 and x = H correspond to the cold and hot matrix sides accordingly, the parameter b is expressed through the specific mass consumption of the gas mixture G = ρ_{0}U_{0}, its specific heat and effective heat conductivity of

the porous matrix

The given theory satisfactorily described the process of surface burning on ceramic permeable matrices with regular structure and on some foam metal matrices with low surface permeability [_{g} = η_{s}.

At last time, the technology of metal foam production was developed so that matrices with high porosity and different topography of pores appeared [_{gH} instead of the surface matrix temperature T_{sH} in the expression for burning rate (1):

where the temperature coefficient is

The model described above should be updated thus that the own matrix structure could be taken into consideration because dependence of the parameter η_{g} from gas speed can be possible. The matrix parameters appear through the value T_{sH} in the two-temperature model, first of all, through the volumetric heat exchange coefficient between solid and gas phases, volumetric matrix porosity and also through the coefficient of radiation heat conductivity which is essential for high porosity matrices [

for small Reynolds numbers Re = 2 - 30 [

ductivity can be determined from expression

cording [

The temperature profiles in solid and gas phases inside the matrix can be found from solution of two heat transfer equations for a matrix bulk and gas:

where

tive thermal conductivity of porous matrix without taking into account of radiation transfer is

Boundary conditions: the gas temperatures T_{g} = T_{0} and

on the back (cold) side of the matrix (x = 0) is determined by radiation losses. At x = H the temperature of the working surface of the matrix T_{sH} is given by the value which must be found from solution of the energy balance equation for gas above the matrix surface.

The solution of the system (5) can be reduced to the solution of single equation of the second power:

where

The expression for derivative y(x) and function

Here, _{1}, С_{2}, С_{3} are found from boundary conditions:

Solution (8) includes unknown value T_{s}_{0}, which one can be found by differentiation (7) and inserting the result obtained into the first equation of the system (5) at x = 0:

The gas temperature distribution in the matrix body is obtained from solution of the second equation of the system (5):

Substituting the expressions (8) in (9), we have finally:

The heat balance equation above the working surface of the matrix is given as:

Here, dependence of the specific heat of combustion products on the temperature and occurrence of radiation losses from the back (cold) side of the matrix were taken into account. The effective specific heat of the mixture c_{e} can be

found from the relation

combustible for methane-air mixture is_{p}(T_{f}) can be represented as a linear function from the temperature _{p}_{0} ≈ c_{0} for simplification of calculations with adequate accuracy. Heat

losses coefficient for the backside of the matrix in (11) is

Basically, the parameter n in expression (4) is a function of the adiabatic combustion temperature T_{a}, i.e. n = n(T_{a}). The expression (1) satisfactorily describes experimental data for methane-air mixture [_{a}) is chosen as

Solution of the problem can be found with using (4), (8), (10), (11) and condition of flame front stationarity, i.e. equality of both speeds of the flame front and mixture flowing out from matrix pores. There is some peculiarity of metal foam having cells and channels of different size. The channels of small diameter penetrate in the large cells. It can be assumed that if flame front is located near the matrix surface its speed is equal to the speed of gas streams, which have size of

the large cell diameter, i.e.

front penetrates into the near-surface cells its speed is equal to the speed of gas streams, which have size of the channel diameter. In common case this peculiarity can be taken into account by entering the correction function η_{g} = ψ(U_{0}) in the expression for flame front stationarity which determines a geometry of the front:

For simplicity, the linear empirical function ψ from speed U_{0} can be chosen as:

Here, ψ(U_{0}) is actually equal to the surface matrix permeability in the speed range_{s}_{1} and η_{s}_{2} are calculated for chosen matrix. Function ψ(U_{0}) is equal the total relative area of large cells on the matrix surface η_{s}_{2} ≡ η_{s} at the mixture speed U_{02} which can be chosen as the critical value when the burning failure occurs. In this case, the value U_{02} is found from solution of the problem. By the way it can be taken from experiment. In the other limit case ψ(U_{0}) is equal the total relative area η_{s}_{1} of small channels into the matrix cells at the mixture speed U_{01} which can be chosen as zero.

The matrix surface temperature can be found as a function from the flame temperature which one in turn is expressed through the initial mixture speed U_{0} (4, 12):

Here, the parameter of the problem is_{f}) is ex-

pressed through the function_{s}(T_{f}), radiation losses coefficient K_{l}(T_{f}) and through the emitting coefficient of the matrix surface K_{e}, which generally depends on the matrix surface tempera-

ture_{f})

can be found from solution of the equation system (5). It can be noted that

the mixture speed and it is equal

Parameters corresponding to the experimental conditions were used in our calculations. Main specifications of the solid phase: H = 8 and 14 mm, characteristic cell size d = 0.4 mm, pores per inch of 60 PPI, η = 0.9, η_{s}_{1} ≈ 0.1 η_{s}_{2} ≈ 0.33, thermal conductivity of Chromal λ_{0} = 20 Wm^{−1}・K^{−1}, ε = 0.9 [

The surface and flame temperatures of the matrix as a function of firing rate is introduced in _{0}) (13). Theoretical results (limit case when the flame front area is equal to the matrix surface area) at ψ = 1 were compared with experimental data [^{2}. The critical value w of flameout was also too large w = 162 W/cm^{2} (

Reducing the parameter ψ up to 0.23 decreases the temperature T_{sH}, however, dramatically reduces the critical value of firing rate w = 41 W/см^{2} (_{s} and K_{l} are predetermined by solution of the system (5). The indicated contradiction can be overcome only by assuming dependence of the parameter ψ on the mixture speed i.e. from the specific combustion power.

The solution (14) with expression ψ = ψ(U_{0}) (13) most adequately describes experiment [_{s}_{0}(w) on the firing rate correlate with T_{sH}(w) (_{sH}. The backside temperature of the matrix reduces because of intensive cooling of the matrix with the incoming cold mixture.

Comparative distributions of the temperature in solid and gas phases of the matrix of 8 mm thickness for variable ψ and different firing rate are displayed in _{s} > T_{g} for all considered variants. The more the surface temperature of the matrix and, therefore, the heat flux into the matrix the more the temperature difference. Note, the temperature coefficient is closed to unit even for ψ = 1 (flat geometry of the flame front). For example, Ks = 0.84 at firing rate w = 40 W/cm^{2}. Dependence of the coefficient K_{s} on firing rate for variable geometry of the flame front is presented in

Radiation losses from the backside of the matrix grow when T_{s}_{0} approaches to T_{sH} at reducing firing rate. They are approximately 10% in the area of w ~40 W/cm^{2}.

The similar result was obtained for thicker matrix of 14 mm thickness at variable geometry of the flame front (_{s} = 0.94 at w = 20 W/cm^{2} and K_{s} = 0.88 at w = 40 W/cm^{2}.

The suggested approach can be useful for analysis of the surface burning on matrices of different structure and material including with surface coating. It allows us to explain some particularities of the surface burning.

The matrices from metal foam (Chromal) of 8 mm thickness, with volumetric porosity about 0.9, pore density of 60 PPI were used in the study. An elemental composition of metal: Cr-18%; Al-6.5%; Co-1.5%; Fe-basic. Some matrices were coated with ceramic film (alumina) of the thickness ~20 μm (

The experimental studies were carried out using the model burner device with a removable plate matrix (

The mixture of natural gas with air was formed in the mixer and flowed to the burner device. The ratio between components could be varied in the wide range.

That provided changing the air excess coefficient from 0.4 up to 2, however, all experiments were conducted at α = 1.05. Gas and air consumptions were measured and regulated by F201AV and MV-304 flow meters (Bronkhorst High-Tech, USA) accordingly. Variations of the environmental temperature and pressure were not considered because of negligible in comparison of variation of nature gas pressure in the pipeline (~3%). The radiation pyrometer AR-882 IR (HM Digital Ltd., USA) with working wave range of 8 - 14 μm was used to measure the radiation temperature of the matrix surface averaged over a surface area of ~1 cm^{2} within the central part of the matrix. The temperature of the backside of the matrix was measured by alumel-chromel thermocouple of 0.3 mm thickness. The integral intensity of the radiation flux from the matrix surface was measured by pyrometric sensors IRA710ST1 and IRA-E420S1 in the spectrum from visible up to 14 μm and from 5 up to 14 μm accordingly. Record of the signals from the thermocouple and pyrometric sensors with using electronic converter Е-270 (l-Card Ltd. Russia) was executed on the computer. The dispersion filters F1 and F2 in the narrow spectrum range of 1 - 3 μm and 6 - 7 μm accordingly were used for measurements of the spectral radiation flux. The filter F1 with trapezoidal transmittance had a transmittance plate in the wave range of 1.2 - 2.8 μm and attenuation of 0.5 in wavelengths of 1 and 3 μm. The filter F2 with a bell-shaped transmittance had transmittance maximum in 6.6 μm and attenuation of 0.5 in wavelengths of 6.2 and 7.2 μm.

The sensors were fixed aside from the burner at the distance of 400 mm from the matrix center at bevel way of 45˚ to the horizontal (position A in ^{2} [

The experiments were carried out with both matrices without and with coating at different specific firing rate, which was varied from ~20 up to 60 - 80 W/cm^{2}. Damping or breaking-off of the flame occurred on the firing rate boundaries accordingly. The radiation temperature, radiation flux from the working surface of the matrix and also the backside temperature were measured in each experiment. The visible radiation from the surface of the matrix without coating was brighter than from matrix with ceramic coating. However, the temperature of the surface with ceramic coating measured by the IR pyrometer was above approximately on 100 - 200 K in all ranges of changing parameter w (

the working surface temperature. The thermocouple showed higher temperatures for the matrix with coating (

The experimental results obtained were compared with calculations (

Comparison of experimental and computed results for both matrices is enough satisfactory taking into account simplicity of the model and some uncertainty in physical data for the matrices. Note that the temperature dependence for the working surface of the matrix on the parameter w is smoothly, and the backside temperature is much lower for matrices of 14 mm thickness [

Radiation fluxes from the surface of both matrices without and with ceramic coating in wide and narrow (

Here, the constants

It was appeared that the radiation flux was almost twice above for the matrix with ceramic coating in comparison with the matrix without any coating in the spectral range λ > 5 μm at all firing rate (

as Chromal for matrix without any coating reduces sharply up to value about 0.4 - 0.5 with increasing wavelength from 5 up to 12 μm [

The opposite effect is watched in the spectral range 6.2 < λ < 7.2 μm where spectral transparency window for ceramics is absent (

Comparison of integral radiation fluxes for both matrices demonstrates that the relation of experimental and estimated radiation flux intensities K is close to unit in the wide range of firing rate 10 > w > 50 W/cm^{2} (

K becomes notably more unit (up to 1.5 times) at w > 50 W/cm^{2}. It is connected with growing of the difference in the surface temperatures at increasing firing rate. .

Calculated values K with using computed and experimental (

Thus, a metal foam matrix with ceramic coating is not worse in radiation efficiency than a matrix without any coating but noticeably surpasses it at specific firing rate w > 50 W/cm^{2}. Apparently, this fact is determined essentially with thickness of the ceramic coating. The integral surface emissivity will grow with reducing thickness of the ceramic film. However, the temperature of the matrix surface can be dropped.

The comparative analysis of the thermal and radiation characteristics of the surface burning on metal foam matrices was carried out. The modified two-tem- perature model for analytical finding the temperature distribution in gas and solid phases into the matrix body and the flame temperature was offered. It was shown that topographic particularities determining the heat exchange processes in the matrix body and the surface interaction of gas streams with the flame front have to be taken into account for metal foam matrixes. The analysis of radiation fluxes in different spectral regions with using of pyrometric sensors and dispersion filters was executed. Comparison of radiation properties of both matrices without and with ceramic coating was done. It was found that the integral radiation efficiency of the matrix with ceramic coating at surface burning was comparable with radiation efficiency of the matrix without any coating in the wide range of the firing rate and surpassed it on 30% - 40% at firing rate w > 50 W/cm^{2}. The radiation efficiency of the matrix with ceramic coating is higher up to 2 times in the spectral range from 5 up to 14 μm.

Shmelev, V. (2017) Radiation Efficiency of Surface Burning on a Foam Metal Matrix with Ceramic Coating. Energy and Power Engineering, 9, 366- 385. https://doi.org/10.4236/epe.2017.97025

A pre-exponential factor (m/s)

a_{1} parameter (m^{−2})

a_{2} parameter (m^{−1})

a_{3} parameter (m^{−1}・K^{−3})

B parameter (K^{−2})

B_{1} parameter (K^{n−4})

b parameter (m^{−1})

C_{1} constant (Km^{−1})

C_{2} constant (Km^{−1})

C_{3} constant (K)

c_{0} specific heat of the input mixture at constant pressure (J・kg^{−1}・K^{−1}) and temperature T_{0}_{.}

c_{e} effective specific heat of mixture at constant pressure (J・kg^{−1}・K^{−1})

c_{p} specific heat of combustion products at constant pressure (J・kg^{−1}・K^{−1})

c_{p}_{0} specific heat of combustion products at constant pressure (J・kg^{−1}・K^{−1}) and temperature T_{0}

d diameter of cells (m)

E activation energy (J・mol^{−1})

F function

G specific mass consumption of mixture (kg・s^{−1}・m^{−2})

H thickness of the matrix (m)

J radiation flux (W・m^{−2})

K relation of radiation fluxes

K_{e} emitting coefficient

K_{l} heat losses coefficient

K_{s} temperature coefficient

k_{e} extension coefficient (m^{−1})

k coefficient in equations

n power in expression for flame rate

Nu Nusselt number

Pr Prandtl number

Q-combustion energy of unit mass of combustible (J・kg^{−1})

R universal gas constant (J・mol^{−1}・K^{−1})

Re Reynolds number

s specific surface of porous layer (m^{−1})

T_{0} initial gas temperature (K)

T_{a} adiabatic temperature of combustion (K)

T_{f} temperature of the flame front (K)

T_{g} gas temperature in matrix body (K)

T_{gH} outlet gas temperature above working surface of matrix (K)

T_{s} temperature of solid phase in matrix body (K)

T_{s}_{0} temperature of the cold surface (solid phase) of matrix (K)

T_{sH} temperature of the hot working surface (solid phase) of matrix (K)

U_{0} input gas speed (m/s)

U_{f} flame front speed (m/s)

U_{g} gas speed above matrix surface (m/s)

w firing rate (Wcm^{−2})

x coordinate in matrix body (m)

α air excess coefficient

α_{v} volumetric_{ }heat exchange coefficient (W・m^{−3}・K^{−1})

β coefficient of proportionality (J・kg^{−1}・K^{−2})

ε working surface emissivity

η matrix porosity

η_{g} relative cross section of_{ }gas streams

η_{s} surface_{ }permeability of matrix

λ_{0} thermal conductivity of matrix material (W・m^{−1}・K^{−1}).

λ_{*} effective thermal conductivity of porous matrix (W・m^{−1}・K^{−1})

λ_{g} thermal conductivity of mixture (W・m^{−1}・K^{−1})

λ_{m} total thermal conductivity of matrix (W・m^{−1}・K^{−1})

λ_{r} coefficient of radiation conductivity (W・m^{−1}・K^{−1})

λ wave lengths (µm)

ξ_{m} concentration of combustible

ρ_{0} density of the input mixture (kg/m^{3})

ρ_{m} density of combustible (kg/m^{3})

σ Stefan-Boltzmann constant (Wm^{−2}・K^{−4})

ψ function of surface flame correction

1; 2; 3 index

a adiabatic

f flame front

g gas

H surface coordinate

m methane

r radiation