4.1.1 Mass Balance

The two components in this problem are polymer and gas, both of which are treated as fluids. The transformation of the polymer from solid to melt as the temperature rises above the glass temperature is handled through changes in the polymer viscosity. The densities of polymer and gas are defined locally as ρ_p = M_p/V_p and ρ_g = M_g /V_g respectively, where M_p is the mass of small volume Vp of polymer and Mg is the mass of small volume V_g of gas. The local volume fractions of polymer and gas are φ_p = V_p/V and φ_g = V_g /, where = V_p + V_g is the total volume. The sum of local volume fractions is equal to one, and the initial conditions are φ_p =1 and φ_g = 0.

Mass is exchanged between the two components of polymer and gas as the thermoplastic sample is heated. Because conservation of mass requires that the total mass remain constant, whatever is lost from the polymer must be gained by the gas and vice versa. For many polymers, including those considered in this report, the rate of mass loss as function of temperature T is adequately described by first order Arrhenius expression,

= ρpφpB exp(−E/RT)   ,

(1)

where B and E are the pre-exponential constant and activation energy respectively for the gasification reaction and R is the universal gas constant [25]. Therefore, assuming that all variables and material properties depend on radius alone and act in the radial direction only, the mass balance equations for the polymer and gas components in spherical geometry are

(2)

(3)

where W_p is the radial velocity of the polymer and W_g is the radial velocity of the gas.

The sum of these two equations demonstrates the conservation of total mass,

(4)

where the total mass density is

(5)

and the barycentric velocity is

(6)