5.9 Component Velocities

The final quantity required to solve the energy equation (7) at the end of this timestep is the barycentric velocity W*, defined by equation (6). This variable can be determined directly from the total mass conservation equation (4). Similar calculations can be used to determine the individual velocities of polymer and gas components, W_p and W_g.

The finite element approach requires consideration of jump conditions between elements. This arises from the fact that volume fractions are uniform throughout an element and may change considerably from element to element, while material properties such as density and specific heat, often functions of temperature, are continuous across element boundaries.

To determine the barycentric velocity for element i,

(47)

consider the total mass conservation equation within the element,

(48)

The density , given by equation (44), is continuous within the element but jumps in value from one element to the next.

Since the node locations r_i and r_i+1 bounding element i change with time, introduce a linear local coordinate system:

(49)

(50)

where L_i(t) = (r_i+1 − r_i) is the thickness of the element. This converts nodes at r_i and r_i+1 to fixed nodes at x = −1 and 1 respectively for the element under consideration. Replacing the derivatives with

where W_i = dr_i /dt and W_i+1 = dr_i+1/dt are velocities of inner and outer nodes respectively, the mass conservation equation can be rewritten as

(53)

The nodal velocities are calculated from the node locations at this and the previous timestep as

(54)

where superscripts k and k + 1 denote successive timesteps. As defined in equation (52), velocity W within the element is interpolated linearly from the nodal velocities. Integrating this equation within a single element from x = −1 to x = 1 results in an equation for baryclinic velocity at the outer node given the velocity at the inner node:

(55)

where the subscript (i + 1)− indicates a value just within element i from its outer node and (i)+ indicates just within from the inner node. For the simplifying case in which polymer and gas densities are considered constant, this equation becomes:

(56)

with = ρ_pφ_pi + ρ_gφ_gi uniform throughout the element.

The jump condition in baryclinic velocity from one element to the next is determined by integrating equation (53) carefully from r_i+1 − Δ in element i to r_i+1 + Δ in element i + 1 and taking the limit Δ → 0. Note that the first two terms in the equation jump in value to the next element without the presence of singularities at the node. The integrated values of these two terms over a narrow region surrounding r_i+1 therefore goes to zero as the size of the region goes to zero, and the jump condition is

(57)

The product ρ*(W* − W_i) is thus a nodal quantity, although both total density ρ* and W* jump from one element to the next. This equation is simply an expression for the conservation of mass flux across element boundaries, in a frame of reference moving with the boundary. It is analogous to the conservation of heat flux across boundaries,

(58)

which is automatically enforced by the finite element method, as discussed in Appendix B.

The velocities of the individual components may be determined in a similar way from mass balance equations (2) and (3) for polymer and gas respectively. The resulting equations include a term to account for mass lost or gained. For the simple case in which material properties are assumed constant in time and space, the polymer velocities are given by

(59)

(60)

and gas velocities by

Note that even in the frequently investigated limit in which the gas is assumed to escape instantaneously, resulting in volume fractions of φ_g = 0 and φ_p = 1 throughout the sample, the product of φ_gW_g (given infinite gas velocity) is finite and contributes to the convection of heat in the energy equation. The consecutive determination of component velocities is begun by fixing the center of the spherical sample, so that W_p1 = W_g1 = = 0.