1.2 Fluid Mass Fraction Conservation

The mass fraction of each fluid, or its "presence probability", mk, 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(rmk)/dt+ div(rVmk- Gt grad mk ) = Rm,k

Rm,k, the net rate of k-fluid generation, is the balance of micromixing rate, Rmix,k, and interphase transfer, Sp,k:

Rm,k = Rmix,k + Sp,k

The source term, Sp,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, Rmix,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:

Rmix,k = r Si Sj Fk,i,j mimj Ti,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;
• mi and mj are the mass fractions of fluids i and j;
• Ti,j measuring the turbulent motion, has dimensions 1/s and
• Fk,i,j is the fraction of mass lost in the ij encounter which enters fluid k.
For all the computations reported below, Ti,j is assumed to be independent of i and j; and calculated, inversely proportional to the eddy-break-up time scale, as:

Ti,j = Cmixe/K

with K standing for the kinetic energy of turbulence, e for its dissipation rate and Cmix 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.
```

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:

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