Each governing differential equation is discretized into a set of algebraic equations that are solved using the tri-diagonal matrix algorithm. All differential equations for conservation of mass, momentum, energy, and (when appropriate) for solvent in the fiber are solved sequentially (that is, segregated from one another). Because the governing equations are coupled and nonlinear, several iterations have to be performed to obtain a converged solution. The solution process consists of several steps outlined below:
The fiber properties are updated based on the initialized or the current solution.
The friction factors for momentum exchange between the fibers and surrounding fluid are computed based on current values of fiber and fluid velocities.
The fiber momentum equation is solved and current values of the mass fluxes in the fiber are used.
The heat transfer coefficients are computed using Reynolds numbers from the beginning of the iteration loop.
The fiber energy equation is solved.
In the case of dry spun fibers, the equation for the mass fraction of the solvent is solved. First, the mass transfer coefficient is updated. The evaporated (condensed) mass is computed based on the vapor liquid equilibrium at the beginning of the iteration loop. Finally, the governing equation is solved.
The mass fluxes and the diameter of the fiber cells are updated.
A check for convergence of the equation set is made.
These steps are continued until the convergence criteria are met for all equations of the considered fiber or until the number of iterations exceed the given limit.
This solution algorithm is applied to all defined fibers.