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_im_j T_i,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;
• m_i and m_j are the mass fractions of fluids i and j;
• T_i,j measuring the turbulent motion, has dimensions 1/s and
• F_k,i,j is the fraction of mass lost in the ij encounter which enters fluid k.

T_i,j = C_mixe/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:

       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:

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