In what follows the computational nature of the model, in terms of conservation equations, will be described in details. It is done in the manner related to the application of MFM-CVA to the turbulent combustion [1].

Then the concluding remarks are drawn and the references are provided.

1 The computational nature of the model;

The combustible flow is considered as a population of different interspersed fluids characterised by their mass fractions and associated attributes.

The former are, always, supposed to obey their own conservation equations. Two classes of the latter are distinguished in the next sub-section. Then the equations solved are described in details followed by the boundary conditions applied. Finally, the basic formulae are given for the attribute processing to get the population- average flow properties.

1.1 PDA and CVA

It is generally assumed here that, in the devices under consideration, the fuel and the oxidant enter the combustion space at different locations and mix together. As a consequence, the "diffusion-flame" type of combustion is supposed to take place within that space and the mixture fraction, f_mix, is the only PDA, Population-Defining Attribute, associated with each fluid in the population [ 1 ].

PDA is uniformly discretized as if each fluid of the population had its own value of mixture fraction uniquely pre-defined as follows:

f_mix,k = ( k-1 )/( N_fluid - 1 )

where k is a current fluid number and N_fluid stands for the total number of fluids in the population.

The implication of the above is that the lowest-number fluid, k=1, is always fuel-free, f_mix,1 = 0, while the largest-number one represents the pure fuel, f_mix,N =1.

The choice of CVAs depends upon the situation in question. It forms the modeller's options and will be specified in the sections corresponding to each of the cases considered.

The few guidelines as mentioned in [ 1 ] are as follows:

• If heat losses by radiation and/or convection to cold walls, can not be neglected, it is the enthalpy which should be treated as a CVA.
• If the validity of the mixed-is-burned assumption is in doubt, the reacted-fuel and other gas-composition fractions can also be treated as a CVA.

Unlike PDA, the values of CVA are not pre-defined. They are calculated from their own conservation equations as described latter in this section.

1.2 Fluid Mass Fraction Conservation

The mass fraction of each fluid, or its "presence probability", m_k, in a multi-fluid population is assumed to be a conserved quantity. Its value at each point in the flow domain is computed by PHOENICS through solution of the following conservation equations of conventional type:

d(rm_k)/dt+ div(rVm_k- G_t grad m_k ) = R_m,k

R_m,k, the net rate of k-fluid generation, is the balance of micromixing rate, R_mix,k, and interphase transfer, S_p,k:

R_m,k = R_mix,k + S_p,k

The source term, S_p,k, is due solely to transfer of mass into the gas phase from reacting particles (e.g. coal). In all other cases there are no such a source.

The term, R_mix,k, is resulting from micromixing of the fluids as they move past, or collide with, each other in their turbulent motion. It is expressed, for uniformly-divided population, as:

R_mix,k = r S_i S_j F_k,i,j m_im_j T_i,j

wherein:

• Ss indicate the summation for all values of index i or j (and it is immaterial whether summation over i or j occurs first), with attention to i less than j;
• m_i and m_j are the mass fractions of fluids i and j;
• T_i,j measuring the turbulent motion, has dimensions 1/s and
• F_k,i,j is the fraction of mass lost in the ij encounter which enters fluid k.

T_i,j = C_mixe/K

with K standing for the kinetic energy of turbulence, e for its dissipation rate and C_mix for an empirical constant.

The fractional loss of mass is computed by following rules:

       Fk,i,j   = -0.5       for k=i or k=j  and j greater than i+1,
               =  0.0       for k less than i or k greater than j
                            or  j=i+1,
               =  1/(j-i-1) for all other values of i, j and k.

These hypotheses are what is called, in MFM parlance, as Promiscuous-Mendelian coupling/splitting scheme of Brian Spalding [ 2 ].

The sources that resulting from the above scheme applied to the interactions between,say, the 5 fluids with T=r=1,would be in fact as follows:

R_mix,1=-0.5(m₃+ m₄+ m₅)m₁
R_mix,2= m₁m₃+ m₁m₄/2+ m₁m₅/3- 0.5(m₄+ m₅)m₂
R_mix,3= m₄(m₁/2+m₂)+ m₅(m₁/3+m₂/2)- 0.5(m₁+ m₅)m₃
R_mix,4= m₃m₅+ m₂m₅/2+ m₁m₅/3- 0.5(m₁+ m₂)m₄
R_mix,5=-0.5(m₁+ m₂+ m₃)m₅

1.3 Transport Equations for CVAs

Let C_k be the value of a continuously-varying attribute of fluid k. The conservation equation for C_k takes the following general form:

d(rC_k)/dt+ div(rVC_k- G_t grad C_k ) = Rc_k + Sc_k,p

where Rc_k is within-fluid mass rate of creation and depletion of C_k and Sc_k,p is the rate of creation by addition from the dispersed phase, if any.

The net rate of within-fluid generation is given by the balance of the sources resulting from the ij encounters, Rc_mix,k, and the source of CVA due to its in-fluid generation and/or dissipation,Rc_gen,k:

Rc_k = Rc_mix,k + Rc_gen,k

The contributions resulting from micromixing by fluid encounters are written as:

Rc_mix,k= S_iM_k,i(C_i-C_k)

where M_k,i is the micromixing mass transfer which enters the fluid k from fluid i.

It is calculated as the i-related portion of the total mass transfer to each fluid in ij encounters:

S_iM_k,i= rS_i S_j Fc_k,i,j m_im_jT_i,j

where

       Fck,i,j  = 0.0       for k less or equal than i or k greater or
                           equal than j or  j=i+1,
               = 1/(j-i-1) for all other values of i, j and k.

For the 5-fluids population with T=r=1 the resulting sources are, in fact, as follows:

Rc_mix,1 = 0 ;
Rc_mix,2 =  (m₁m₃/2+m₁m₄/4+m₁m₅/6) (C₁-C₂)
                                         +m₁m₃/2  (C₃-C₂)
                                         +m₁m₄/4  (C₄-C₂)
                                         +m₁m₅/6  (C₅-C₂) ;
Rc_mix,3 =                (m₁m₄/4+m₁m₅/6) (C₁-C₃)
                           +(m₂m₄/2+m₂m₅/4) (C₂-C₃)
                           +(m₁m₄/4+m₂m₄/2) (C₄-C₃)
                           +(m₁m₅/6+m₂m₅/4) (C₅-C₃) ;
Rc_mix,4 =                               m₁m₅/6  (C₁-C₄)
                                          +m₂m₅/4  (C₂-C₄)
                                          +m₃m₅/2  (C₃-C₄)
              +(m₃m₅/2+m₂m₅/4+m₁m₅/6) (C₅-C₄) ;

Rc_mix,5 = 0

The generation/dissipation rates, Rc_gen,k, that appear as source terms of CVA are usually problem specific.

For in-fluid chemical reaction, they can be computed from Arrhenius rate expressions, using the eddy dissipation concept or blending of two, as appropriate.

For example, employing the eddy-dissipation model gives the following reaction rate relation for in-fluid mass fraction of unreacted fuel as CVA:

Rc_gen,k = R_fu,k = - Are/K min( C_fu,k, C_ox,k/s ) m_k

where

• A is a model constant;
• s is stoichiometric coefficient of the reaction;
• C_fu,k represents the within-fluid fuel mass fraction;
• C_ox,k/s represents the within-fluid oxidant mass fraction and
• m_k is the mass fraction of fluid in question.

If the heat losses, say, to the cold walls can not be neglected compared with the heat realise through reaction, then the specific gas enthalpy can be treated as CVA, H1_k, of the fluid.

The source term of the within-fluid heat losses to wall can be expressed as:

Rc_gen,k = R_H1,k = S_H1,km_k

where net rates of heat transfer to the wall, S_H1,k, can be readily accounted for via near-wall heat transfer coefficients, either explicitly or in terms of wall functions employed.

More examples of Rc_gen,k formulations will be shown later in the sections dealing with the case studies.

1.4 Boundary Conditions for m_k and CVA

For the systems considered here the boundaries will be of three general types: walls, fuel/oxidant supply inlets, including distributed ones (e.g., interphase transfer from dispersed phase), and outlets. The following boundary conditions are generally applied:

• At the walls: A zero-flux conditions are applied for all m_k. A wall-function type, fixed-flux or user-defined near-wall conditions are applied for CVA, as appropriate.
• At the outlets: All fluids are asssumed to exhaust into region of uniform pressure.
• At the inlets:All fluids are injected at a given flow rates and at a given fluxes per unit mass.

  For the m_k's conservation equations,the unity values of either m₁ or m_N with zero values for the rest of the population are assumed to control the latter at the oxidant and fuel supply regions, correspondingly.

  The appropriate values of CVAs are specified as consistent with their nature.

Often the fluid attributes, either PDA or CVA, require further processing to get the number of derivatives needed to complete the simulation. The examples of attribute processing are:

• Getting population-average quantities;
• Getting RMS of property fluctuations;
• Computing of auxiliary attributes.

fª_mix= S_km_kf_mix,k

where m_k are used as the weighting coefficients to determine the fluid-averaged mean values.

That of a CVA, such as C_k, is given by similar relation:

Cª_k= S_km_kC_k

The RMS of the fluctuations for any attribute, f , can be written as

RMS_f = (S_k (fª_mix-f _mix,k)²m_k)½

The example of auxiliary attribute is a fluid temperature, T_k. It can be calculated via two within-fluid CVAs, namely, total enthalpy, H1_k, and mass fraction of fuel, C_fu,k as follows:

T_k = H1_k - C_fu,kH°

wherein H° represents the specific heat of fuel combustion.

In some cases, e.g. non-adiabatic "mixed-is-burned" combustion, T_k manifests itself as a joint function of both PDA, f_mix,k, and CVA, H1_k, whereas for the adiabatic infinitely-fast-reaction combustion system it becomes the function of mixture fraction PDA only.

2. Concluding remarks;

This document has supplied the complete statement of the equations solved in MFM-CVA. With this information in hands, it would, in principle, be possible for a PHOENICS user to verify, by means of PLANT or IN-FORM settings, that the fields produced by built-in PHOENICS implementation of MFM for fully-converged solution of identical problems do satisfy the conservation equations listed.

3. References;

1. D.B. Spalding, "Multi-fluid models for simulating turbulent combustion", - Presentation at CODE Annual SEMINAR in Teraelahti, Finland, 3-4 October 2001