22.2. Governing Equations of Fiber Flow

Mass conservation of a fiber element is written as

(22–1)

where

	= fiber density
	= fiber velocity vector
	= surface area vector of the fiber surface parallel to the flow direction
	= mass fraction of the solvent in the fiber
	= evaporated mass flow rate of the solvent
	= fiber diameter
	= coordinate along the fiber

is calculated using a film theory.

(22–2)

where

	= mass transfer coefficient estimated from an appropriate correlation (see Correlations for Momentum, Heat and Mass Transfer)
	= solvent’s molecular weight
	= molar concentration of the surrounding gas
	= mole fraction of the solvent vapor in the surrounding gas

At the fiber surface, the mole fraction of the solvent in the gas is related to the solvent mass fraction in the fiber by the vapor-liquid equilibrium equation given by Flory [185],

(22–3)

where

	= Flory-Huggins parameter
	= absolute pressure in the surrounding flow
	= saturation vapor pressure of the solvent

These equations are used only when dry spun fibers have been selected.

The formation of fibers is based on tensile forces in the fiber that are applied at the take-up point and result in the drawing and elongation of the fiber.

A force balance for a differential fiber element gives the equation of change of momentum in the fiber.

(22–4)

The tensile force in the fiber changes due to acceleration of the fiber, friction force with the surrounding gas, and the gravitational forces.

The friction force with the surrounding gas is computed by

(22–5)

where

	= gas density
	= axial friction factor parallel to the fiber
	= gas velocity parallel to the fiber

The gravitational force is computed from

(22–6)

where is the direction vector of the fiber element.

The tensile force is related to the components of the stress tensor by

(22–7)

Neglecting visco-elastic effects and assuming Newtonian flow one can obtain

(22–8)

(22–9)

leading to

(22–10)

The elongational viscosity is estimated by multiplying the zero shear viscosity by three.

The transport of enthalpy in and to a differential fiber element is balanced to calculate the fiber temperature along the spinning line.

(22–11)

where

	= fiber enthalpy
	= fiber thermal conductivity
	= fiber temperature
	= enthalpy of the solvent vapor
	= heat transfer coefficient

In the case of a melt spinning process, is zero because there is no mass transfer. The term for heat generation due to viscous heating is derived from the fluid mechanics of cylindrical flow to be

(22–12)

Radiation heat exchange is considered by the last two terms

(22–13)

(22–14)

where

	= thermal irradiation
	= fiber’s emissivity
	= Boltzman constant

The fiber enthalpy is related to the fiber temperature as follows

(22–15)

It uses the specific heat capacity of the polymer and the specific heat capacity of the solvent in the fiber.

The enthalpy of the solvent vapor at a given temperature depends on the heat of vaporization , given at the vaporization temperature , and is computed from

(22–16)

where

	= specific heat capacity of the solvent liquid
	= specific heat capacity of the solvent vapor