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_{i}m_{j} T_{i,j}
wherein:
T_{i,j} = C_{mix}e/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:
F_{k,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:
R_{mix,1}=-0.5(m_{3}+
m_{4}+
m_{5})m_{1}
R_{mix,2}= m_{1}m_{3}+
m_{1}m_{4}/2+
m_{1}m_{5}/3-
0.5(m_{4}+
m_{5})m_{2}
R_{mix,3}= m_{4}(m_{1}/2+m_{2})+
m_{5}(m_{1}/3+m_{2}/2)-
0.5(m_{1}+
m_{5})m_{3}
R_{mix,4}=
m_{3}m_{5}+
m_{2}m_{5}/2+
m_{1}m_{5}/3-
0.5(m_{1}+
m_{2})m_{4}
R_{mix,5}=-0.5(m_{1}+
m_{2}+
m_{3})m_{5}