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 in the latter sections.