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_{i}M_{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_{i}M_{k,i}=
rS_{i}
S_{j}
Fc_{k,i,j}
m_{i}m_{j}T_{i,j}

where

Fc_{k,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} =
(m_{1}m_{3}/2+m_{1}m_{4}/4+m_{1}m_{5}/6)
(C_{1}-C_{2})

+m_{1}m_{3}/2
(C_{3}-C_{2})

+m_{1}m_{4}/4
(C_{4}-C_{2})

+m_{1}m_{5}/6
(C_{5}-C_{2}) ;

Rc_{mix,3} =
(m_{1}m_{4}/4+m_{1}m_{5}/6)
(C_{1}-C_{3})

+(m_{2}m_{4}/2+m_{2}m_{5}/4)
(C_{2}-C_{3})

+(m_{1}m_{4}/4+m_{2}m_{4}/2)
(C_{4}-C_{3})

+(m_{1}m_{5}/6+m_{2}m_{5}/4)
(C_{5}-C_{3}) ;

Rc_{mix,4} =
m_{1}m_{5}/6
(C_{1}-C_{4})

+m_{2}m_{5}/4
(C_{2}-C_{4})

+m_{3}m_{5}/2
(C_{3}-C_{4})

+(m_{3}m_{5}/2+m_{2}m_{5}/4+m_{1}m_{5}/6)
(C_{5}-C_{4}) ;

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,k}m_{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.