DETERMINATION OF THE MEAN BEAM LENGTH OF THE COMBUSTION CHAMBER

The evaluation of the elemental partial pressures was based upon Dalton’s law and the measured gaseous concentrations. These pressures were combined with the mean beam lengths of the radiation exchanges among the chamber wall surfaces in order to evaluate the free parameter known as “the wave length” p x L (atm x ft). 

The prediction of the average spatial distribution of the gas temperatures inside the chamber space and the temperature and heat-flux distribution on the walls of the combustion chamber whose geometry and size as well as the enclosed gas properties are specified is presented in the following analysis:

It was assumed that the gaseous mixture inside the chamber was equivalent to a mixture of two grey gases (Carbon Dioxide (CO₂) and water steam (H₂O)) as well as a neutral gas (Nitrogen (N₂)). The total emissivity of this gaseous mixture may be evaluated by the following relationship:  

Were

e_gi=emissivity

a_gi=absorption coefficient

K_i =attenuation coefficient

L =mean beam length

The emissivity for each of the active exhaust gases i (water vapor (i=1) and carbon dioxide (i=2)) was taken from the data provided in Reference (Hottel, 1967) as functions of the corresponding values of the wavelength parameters p_wL_m and p_co2L_m.

The actual in situ measurements of the gaseous and the wall temperatures were conducted by a specialized company (ENCO LTD), which provided as well the gaseous concentrations and the corresponding partial pressures for the running conditions of the facility. As reported above, these data provided the basis for the evaluation of the free parameters (p_iL) needed for the final evaluation of the gaseous emissivity for an assumed constant gaseous temperature equal to 750^oC inside the entire chamber. The emissivity evaluation was based on the experimental data provided by the relevant charts in .11 (page 232), Fig 6.9 (page 229), Fig. 6.12 (page 233) of Reference (Hottel, 1967), with the constant temperature of the enclosed gases inside the chamber transformed into the Rankine scale, i.e. T=1841^oR_a.Actually, the emissivity parameter was evaluated by the following relationship:

whereΔε = The wavelength overlap parameter

These charts apply for a chamber pressurized to a pressure equal to that of the surrounding atmosphere (i.e., 1 atm). When the chamber pressure differs from the atmospheric (in the plant under consideration it was regulated so that it was maintained steady at a magnitude of 0.4 atm) it was necessary to readjust the emissivities determined above by introducing correction coefficients C_{H 2 O} and C_{CO 2}, which multiplied the corresponding two emissivities.

The determination of the correction factors C_{H 2 O} and C_{CO 2} was based on experimental data provided in charts (Fig.13.56 and Fig13.57 in Rathore, Kapuno (, 2011)).

Τhe gas absorptivity and emissivity of the entire mixture were evaluated in a similar manner as the weighted sum of the corresponding contributions for each particular grey gas, i.e.

Were

ε_g=emissivity

a_g=coefficient of gas absorption

a_s=coefficient of surface absorption

The emissivity-pL relationship for the gaseous mixture may be depicted as the weighted sum of the particular contributions of the gray gases participating in it and this relationship can be expressed as

In other words,ε_g is an increasing function of pL with an upper limit unity for a large value of the pL parameter. In this limit

Hence all ε_g values ​​are positive.

For k = 0 there exists no active grey gas component inside the mixture. For non-zero values of k, it may be shown that

Defining the pL_m and 2pL_m values ​​in the above equation, results into the following relations

by replacing the product k_ip = K_i

For the chamber under consideration, K₁becomes

For wavelength of a double magnitude

The mean beam length was defined by H. Hottel (Hottel, 1967)to represent the radius of an equivalent hemisphere so that the incident radiation flow at the center of the hemispherical base equals the average radiation flux incident to the surfaces surrounding the gas volume. According to this Reference this flux may be given by (Fig. 7.1, page 257) H. Hottel (Hottel, 1967))

CALCULATION EMISSIONS, OF WATER VAPOR AND CO₂ AND ITS MIXTURE IN THE COMBUSTION CHAMBER AND CALCULATION OF ABSORPTION COEFFICIENTS

Dalton's law gives the partial pressures and multiplying the mean beam length corresponding to the orbit to obtain the necessary pL parameters for each component of the waste gas mixture to calculate the corresponding CO₂ and water vapor emissions.

We look at Hottel's tables for CO2 and water vapor emission with PL parameter.

The tables are applicable to radiant gases at atmospheric pressure. If they do vary the atmospheric pressure then the corrective factor must be taken into account.

We consider the mixture of constituent gases as a gray gas following Lambert's law so as to reduce the emission calculations in the existing diagrams.

We're counting now

The water vapor emission is calculated from Chart 6-1 and equals

From chart 6.3 become Δε where we have

become the value of ordinates is equal to 0.9.

The parameter value in the curves with a corresponding trajectory is calculated by defining the following product:

Replacing the values ofΔεand the ${\epsilon }_{g_{H_{2}O{,2L}_m}}$ = 0.202 of ${\epsilon }_{g_{C{O}_2{,2L}_m}}$ =0.075 becomes

Because the combustion chamber is at a 0.4 atm pressure, we must multiply each of the terms H₂O and CO₂ by the corresponding correction coefficients resulting from diagrams 8 and 9 (Source Engineering Heat Transfer / Rathore, KapunoFig.13.56, Fig.13.57, Fig.13, respectively, in which the path length is expressed by atm.m rather than atm.ft (as depicted in Fig. 6-10 Hottel) by one atmosphere at a pressure of 0.4 atm.

Further we have

From the diagrams 8 and 9 and taking into account

we obtain

It should be noted that the overall emissivity of the gas mixture is less than the sum of the individual emissions (as if each of the two gases were emitted separately) by a correction factor Δε, because each gas behaves as a duplicate in areas between 2.7 m and 15m. Thus, the total radiation is reduced and this correction factor is drastically reduced at temperatures above 1200^oK. And so, because

From the diagrams depicted on the Figures4και5we obtain

By similar way how calculated K₁we will calculate Κ₂andα_g,2.

And because

From diagrams depicted on the figure8 and 9 because the pressure is 0.4 atm becomes

and so, we obtain

CALCULATION OF ABSORPTION COEFFICIENTS BY LECKNER

Another way to calculate the absorption coefficients with a large approximation is given algebraically by Leckner's relations.

In order to calculate the vapor emissions as well as the CO₂ and therefore the absorption coefficients we will use another algebraic way of calculating the Leckner equation.

Coefficients of the LECKNER equation for water vapor and CO₂

We consider that N₂ has no significant emissivity, so we will calculate according to the model the water vapor and carbon dioxide emissivities.

c_ij( $\frac{T}{1000}$ )ⁱwhereΤmeasuredbyΚ⁰, p_w partial pressure of vapor measured by bar and the wave length L_e by cm. Int his case L_e=7.54ft/3.05ft/m=2.15m=215cmand 1atm=1,01325bar

Calculate now ε_wthe temperature of combustion chamber is 750 ⁰C= 1023 ⁰K

For j=0

For j=1

For j=2

consequently becomes

Calculate now the emissivity ε_{co 2}

For j=1 we have

For j=2 we have

For j=3 we have

we have

Calculate now Δε from the Leckner relations.

From Leck nerrelation we have

We now calculate the final emissivities taking into account the correction factors because the combustion chamber operates at a pressure of 0.4 atm. For these wavelengths.

For the wave length L_m=2.15,2L_m,4Lm corresponding

Replacing the values ε_{g,L m ,} ε_{g,2L m}, ε_{g,4L m}into the relations (15) and (16)

CONFLICTS OF INTEREST/COMPETING INTERESTS

The authors whose names are listed immediately below certify that they have NO affiliations with or involvement in any organization or entity with any financial interest (such as honoraria; educational grants; participation in speakers’ bureaus; membership, employment, consultancies, stock ownership, or other equity interest; and expert testimony or patent-licensing arrangements), or non-financial interest (such as personal or professional relationships, affiliations, knowledge or beliefs) in the subject matter or materials discussed in this manuscript.

AVAILABILITY OF DATA AND MATERIAL

Available after reasonable request.

CODE AVAILABILITY

Not applicable

AUTHORS' CONTRIBUTIONS

Conceptualization: Nicholas Pittas; Methodology: Nicholas Pittas; Formal analysis and investigation: Nicholas Pittas; Writing - original draft preparation: Nicholas Pittas; Writing - review and editing: Nicholas Pittas, Demos P. Georgiou, Vasileios Moutsios, IriniMuravieva; Funding acquisition: Nicholas Pittas; Resources: Nicholas Pittas; Supervision: Demos P. Georgiou.

ETHICS APPROVAL

Not applicable/The suggested paperdoes not enter any ethics issues.

CONSENT TO PARTICIPATE

Nicholas Pittas, Demos P. Georgiou, Vasileios Moutsios and IriniMuravieva have consented to participate in this research study.

CONSENT FOR PUBLICATION

Nicholas Pittas, Demos P. Georgiou, Vasileios Moutsios and IriniMuravieva have consented to publish this research study.