The 7-gases option for combusting gas mixtures

Contents

1. The main idea
2. One or two phases?
3. Thermodynamic equilibrium or finite-rate chemistry?
   1. Equilibrium
   2. Calculating the elemental mass fractions
   3. Calculating the temperature
   4. Finite-rate chemistry
   5. Turbulence-controlled reaction
4. The driving force for inter-phase mass transfer
5. Activation in PHOENICS
6. Library examples

1. The main idea

The purpose

The "seven-gases" option of PHOENICS provides a means of computing the density and temperature of gases involved in single- or two-phase combustion processes; it is useful because it is mainly by way of density and temperature changes that such processes influence velocities, pressure and turbulence.

However, it has a wider significance, as will now be explained.

The seven-gases concept represents an idealization of practically-arising combustion situations, achieved by restricting the number of substances present, and their possible interactions.

The elements

Nitrogen (N) is also present, as an inert element; and others, such as silicon (Si) which may be present in ash, are also treated as not engaging in chemical reactions.

The gases

1. O2, i.e. oxygen,
2. CO2, i.e. carbon dioxide,
3. CO, i.e. carbon monoxide,
4. H2, i.e. hydrogen, and
5. H2O, i.e. water vapour.
6. CxHy, i.e. a hydrocarbon gas, the "volatiles" of coal or wood , or a vaporised oil, the composition of which does not need to be everywhere the same,
7. N2, i.e. nitrogen,

The first six may all engage in chemical reactions, i.e. may exchange chemical elements between them.

Nitrogen too may react chemically, as is well-known to be of practical importance. However the extent of the reaction is usually small enough to be neglected in its effects on gas temperature and density.

The condensed-phase materials

1. coal,
2. oil,
3. wood,
4. char, and
5. ash.

2. One or two phases?

The seven-gases option can be used in both single-phase (ONEPHS=T) and two-phase (ONEPHS=F) flow simulations. However, more options are available than are implied by these words; for "PHOENICS phases" and "thermodynamicist's phases" are not identical, as will now be explained.

• The choice ONEPHS=T implies that all the substances at a given point have the same velocity components and the same temperature.

• However, this need not preclude the presence of both gaseous- and condensed-phase material, such as finely-divided carbon (eg smoke), or oil mist, or even pulverized coal.
  In such a case, the density of the mixture is to be calculated from:

  density =

  1 / ( Mgas/SPVOLgas + Msolid/SPVOLsolid )
  

  wherein:

  Mgas =

  mass fraction of gas

  Msolid =

  mass fraction of solid

  SPVOLgas =

  specific volume of gas

  SPVOLgas =

  specific volume of solid,

  which is often small, but cannot be neglected, because it appears as a denominator.

• The choice ONEPHS=F provides for consideration of two sets of velocity components (unless EQUVEL=T), and two temperatures and other properties; and each of these sets is associated with one of the two "PHOENICS phases".

• However it is still possible to make arbitrary allocations of individual materials to one or other of these phases, independently of the phase to which a thermodynamicist would assign them.

  Examples of the choices which could be made are:-

  1. All the condensed-phase materials might be assigned to PHOENICS-phase 2, and the gases to PHOENICS-phase 1.
     This might suit thermodynamicists; but, it would imply that tiny smoke particles and (comparatively) large coal particles would possess the same velocity and temperature at each point, these being different from those of the gases.

  2. Alternatively, smoke, wood, and char (ie the substance resulting after the wood volatiles had been driven off) might be assigned to PHOENICS-phase 1, on the grounds that they were likely to have temperatures and velocities close to those of the gases; then only the larger coal and oil particles would be assigned to PHOENICS-phase 2.

     This is indeed the choice which is represented in the coding supplied in the GXDENS.FOR file described below.

  3. Another possibility would be to assign also oil droplets to the PHOENICS-phase 1, leaving only coal particles in phase 2, on the grounds that their life-time is much shorter than that of coal.

• What choice should be made cannot be universally prescribed; it must be left to the user's intuition and experience.

3. Thermodynamic equilibrium or finite-rate chemistry?

3.1 Equilibrium

It is then possible to deduce the composition of the gas from knowledge of the mass fractions of chemical elements: FO, FC, FH and FN. The following diagram is of assistance.

Figure 1 shows an equilibrium-composition diagram which represents the six reactive gases (nitrogen being therefore excluded), in respect of their elemental oxygen, carbon and hydrogen contents.

It shows three main regions, namely:

1. the red triangle, where only O2, CO2 and H2O have finite concentrations;
2. the green quadilateral, where only CO2, H2O, H2 and CO may be present, and
3. the blue triangle, where the only possible gaseous constituents (apart from nitrogen, which can be present in any region) are H2, CO and hydrocarbons CxHy.

This diagram has been drawn on the presumption that oxidation of any of the fuels proceeds in two stages, viz:

1. to create CO2 and H2O, and thereafter, as more fuel is added,
2. to create CO and H2.

Any particular mixture can be located as a point (say M) on the diagram; and its mass fractions of the three elements, FO, FC and FH, are proportional to the perpendicular distances from the point to the side of the triangle opposite to the vertex associated with the element in question.

The sum (FO + FC + FH) would equal unity, were other materials absent; but in general it equals:
(1 - FN - F_condensed_phase_materials)
the final term being present, of course, only if these materials are being treated as belonging to PHOENICS-phase 1 because they "flow with the gases".

3.2 Calculating the elemental mass fractions

• sources and sinks at inflow and outflow boundaries, i.e. boundary conditions; and
• (in two-phase flows) transfer of mass between phases, i.e. distributed sources.

The within-gas chemical reactions do not give rise to sources, because, though they transfer chemical elements from one mixture component to another, they do not change the mass of any element.

Among the interesting consequences are:

1. When the combustion chamber is supplied with only two input streams, e.g. air and fuel gas CxHy, then:
   • all the mixture states which exist anywhere in the chamber lie somewhere along the straight line between point CxHy and point O on Fig.1.

   • Then the local "fuel/air ratio" is equal to the ratio of line lengths: O_to_M / M_to_CxHy.

   • Further, the "mixture fraction", which is used as a solved-for variable in many SCRS (ie Simple Chemically Reacting System) cases in the PHOENICS Library , varies linearly along that line from 0.0 at O to 1.0 at CxHy.

2. When there are two fuel-supply streams, each having a different carbon-to-hydrogen ratio and therefore being represented by a different CxHy point on the base line of Fig.1, then all mixture states within the combustion chamber are represented by points lying within the triangle having the vertices:
   O, CxHy1 and CxHy2.

3. There may, of course. be several inlet streams;and they may all have different compositions.

   When there are n sources of supply of gas-phase material, it is best to solve for n-1 conservation equations, for variables which might be named: MXF1, MXF2, MXF3; and then to deduce the elemental mass fractions of carbon, oxygen and hydrogen from:

   FC = sum ( FC1*MXF1 + FC2*MXF2 + ........ + FCn*MXFn )
   where:

   • FC1 = elemental carbon content of supply-stream 1;
   • FC2 = elemental carbon content of supply-stream 2, etc;
   • sum ( MXF1 + MXF2 + .......... + MXFn ) = 1.0 ; and
   • transfer of material from a second phase into the first also counts as a supply stream. with similar expressions for FH and FO.

However they are computed, it is the values of FC, FO, and FH which dictate the relative proportions of O2, CO2, CO, H2O, H2, CxHy (treated as one material) and FN in the gaseous equilibrium mixture.

Wood and char

Species mass fractions begin with:

• Y (e.g. YCO2) if they relate to the whole of phase 1 and with
• Z (e.g. ZCO2) if they relate to the gaseous part only.

The diffusion coefficients of the gaseous species are all taken as equal, as are the specific heats; and the reaction rates are diffusion-limited. As a consequence, all species concentrations depend, in piecewise-linear fashion, on the elemental mass fractions.

The values of oxygen fraction FO at which the formulae exbibit discontinuities of slope are called:

• FOPART, where the oxygen has consumed part of the fuel, so as to create CO and H2; and
• FOFULL, where the products of combustion are CO2 and H2O.

• CINCL, the mass fraction of C in the fuel, set via RHO1A or RHO2A;
• HINCL, the mass fraction of H in the fuel, set via RHO1B or RHO2B;
• CINWD, the mass fraction of C in the wood set via SPEDAT
• HINWD, the mass fraction of H in the wood set via SPEDAT
• Also supplied is:
• NINCL, the mass fraction of N in the fuel, set via RHO1C or RHO2C.

3.3 Calculating the temperature

      T = ( H - HCHX*YCHX  - HCOCO2*YCO - HHH2O*YH2 
                          - HCHAR*YCHAR - HVOL*YVOL ) / CP
   

• in the Q1, by the presence of lines such as:
  STORE(YCO,YH2O,...; and
• in the EARTH solver, by non=zero values of:
  LBNAME(YCO), LBNAME(H2O) etc.

The Fortran coding which computes the gas density according to the seven-gases presumption is in gxdens.for, which can be seen by clicking here.

The numerical values which appear in it are:-

• the mass fractions of O2 and N2 in air, namely:
  0.232 and 0.768
• the molecular weights of O2, N2, CO2, CO, H2O, H2, CHx:- (The latter is presumed to be have the same weight as propane)
  32.0, 28.0, 44.0, 28.0, 18.0, 2.0 and 44.0
• the Universal Gas Constant,
  GASCON: 8.31413 E3
• the heat of reaction for C + O2 -> CO2, HCCO2:
  3.279E7
• the heat of reaction for C + 0.5*O2 -> CO , HCCO :
  9.208E6
• the heat of reaction for H2 + 0.5*O2 -> H2O, HHH2O:
  1.209E6
• the specific heat at constant pressure, CP :
  1.100E3

3.4 Finite-rate chemistry

If one or more of the chemical reactions needed to bring about the equilibrium composition represented by Fig. 1 can not reasonably be presumed to proceed with sufficient rapidity, it remains useful to compute the elemental mass fractions; but it is not sufficient.

For example, it might be supposed that the reaction:
2*CO + O2 -> 2*CO2
is the one non-fast reaction.

Then, whereas it might still be presumed that H2 and O2 could not coexist in the red and green regions of Fig.1, CO and O2 could do so.

What values would their concentrations attain? The question could be answered only by solving an additional differential equation, which would have to be provided with a source term representing the finite rate of the CO-oxidation reaction.

It would be immaterial whether the variable solved for were the concentration of CO, of O2, or of CO2; for each can be computed, if one of them is known, from the known values of FO and FC.

Of course, as is well-known, additional gas components such as the "radicles" [O] and [OH], play essential roles in the CO-combustion process; but these are usually such low concentration as not appreciably to affect the density or temperature, which is what the 7-gases option is designed to do.

The subject of how finite chemical reaction rates are to be computed is too extensive to be entered upon here. Attention will be switched instead to the more-frequently encountered reason for departures from equilibrium, namely turbulence.

3.5 Turbulence-controlled combustion

(a) The importance of fluctuations

The time-average mixture in a combustion chamber is often NOT in equilibrium for a reason which is explicable in terms, not of finite-rate chemistry, but of the presence of high-frequency fluctuations of concentration and temperature, i.e. of turbulence.

Precise description of this phenomenon is still beyond the scope of science; but useful idealizations exist, of which one of the most commonly employed is that the gases at any point within a combustion chamber act like a random intermingling of two gases having the same elemental composition, but of which:

• one is in thermodynamic equilibrium, and
• the other is entirely unburned.

It is this concept which underlay the "eddy-break-up (EBU) model", first proposed in 1971. Although EBU has been superseded by the "multi-fluid" (MFM) model, it still allows the major effects of turbulence-chemistry interactions to be reaistically represented.

In MFM terms, EBU represents:

• a "two-fluid" population, with
• "reactedness" as the "population-distinguishing attribute" (PDA), whereby
• one fluid has unity reactedness, and
• the other has zero reactedness.

(b) How to compute the relative proportions of the two fluids

• the product of the concentrations of the fluids participating in an interaction, multiplied by
• some measure of the local turbulent motion, usually taken as proportional to "epsilon / k", the specific rate of decay of turbulence energy, k, when th k-epsilon turbulence model is in use.

Although the details differ, and the EBU formulation is the more direct, both involve the solution of at least one additional equation transport equation having a source term proportional to the above-mentioned product, from which the mass fractions of the two fluids can be deduced.

From these, and from the densities and temperatures of the individual fluids, the mixture-mean densities and temperatures can be deduced.

(c) How to compute the compositions of the two fluids

In relation to the 7-gases model of density and temperature, it is clear that the composition of the first of these fluids is to be calculated as indicated above under the heading "equilibrium".

And the second? Its composition is deducible for a two-supply-stream situation from the local value of MIXF; and specifically the concentration of each species is equal to:

its concentration in the fuel-supply stream * MIXF
+
its concentration in the oxidant-supply stream * (1 - MIXF) .

Once FCMX, FHMX, FOMX and FNMX have been computed in this way, the compositions of the two components of the fluid population can be computed.

Specifically,

• the composition of the burned gas is computed from FCMX, FHMX, etc in the way described above for equilibrium gases, and

• the composition of the unburned gas is computed, species by species, from:

  concentration of species A =
  MXF1 * concentration of A in supply stream 1 + MXF2 * concentration of A in supply stream 2 + ..... MXFn * concentration of A in supply stream n

How to compute the temperatures of the two fluids

4. The driving force for inter-phase mass transfer

5. Activation in PHOENICS

6. Library examples

Input-file library cases which exemplify use of the 7-gases model can be seen by using the PHOENICS Commander's library-search facility

Of these, case 477 is a PIL "macro", loaded into other cases by the load(477) command, or simply #477.

Case 477 itself call another macro, namely case 478, which is where, among other things, the non-PIL variables

CINCL

i.e. mass fraction of carbon in coal

HINCL

i.e. mass fraction of hydrogen in coal

NINCL

i.e. mass fraction of nitrogen in coal

In case 477, they are assigned to the PIL variables RHO1A, RHO1B and RHO1C for transmission to Earth via EARDAT.