1.3 Transport Equations for CVAs

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

d(rCk)/dt+ div(rVCk- Gt grad Ck ) = Rck + Sck,p

where Rck is within-fluid mass rate of creation and depletion of Ck and Sck,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, Rcmix,k, and the source of CVA due to its in-fluid generation and/or dissipation,Rcgen,k:

Rck = Rcmix,k + Rcgen,k

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

Rcmix,k= SiMk,i(Ci-Ck)

where Mk,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:

SiMk,i= rSi Sj Fck,i,j mimjTi,j


       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:

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

Rcmix,5 = 0

The generation/dissipation rates, Rcgen,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:

Rcgen,k = Rfu,k = - Are/K min( Cfu,k, Cox,k/s ) mk


Here, the rate of reaction has been taken as proportional to the "presence probability", mk, so as to preclude the fuel depletion in non-existent fluid.

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, H1k, of the fluid.

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

Rcgen,k = RH1,k = SH1,kmk

where net rates of heat transfer to the wall, SH1,k, can be readily accounted for via near-wall heat transfer coefficients, either explicitly or in terms of wall functions employed.

More examples of Rcgen,k formulations will be shown in the latter sections.