(668g) Transport and Energy Processes in Polymeric Processing

Polymeric materials are used in many forms such as plastics, fibers, rubber, paints, coatings or adhesives and also as molded and fabricated articles implies that there must be a variety of ways in which the compounded resins can be processed and converted into finished products.

Polymers can be classified in different ways, as (i) Natural and synthetic polymers- natural polymers such as cotton, silk, wool and rubber,synthetic PE,PP,etc. (ii) Organic and inorganic polymers- a polymer whose backbone chain is made of carbon atoms is termed an organic polymer and some atoms attached to the side valancies of the backbone carbon atoms may be hydrogen, oxygen, nitrogen, etc. (iii) Thermoplastic and thermo-setting polymers- Polymers may soften on heating, reshaping, retaining the shape and stiffen on cooling and can be repeated several times are termed as thermoplastics e.g., PE, PP, etc. Some polymers, on the other hand undergo some chemical change on heating and convert themselves into an infusible mass, are called thermo-setting polymers e.g., yolk of egg. (iv) Elastomers- when polymers vulcanized into rubbery products exhibiting good strength and elongation, are termed as elastomers e.g., natural rubber, synthetic rubber, silicon rubber. (v) Fibers- polymers drawn into long filament- like materials, whose length is at least 100 times its diameter, are called fibers e.g., nylon and terylene. (vi) Resins- Polymers used as adhesives, coatings, etc., in a liquid form e.g., epoxy adhesives.

Molecular theories for polymeric liquids include the kinetic theories for polymers and can be divided roughly into two classes: network and single- molecule theories:

1. The network theories were originally developed for describing the mechanical properties of rubber. One imagines that the polymer molecules in the rubber are joined chemically during vulcanization. The theories have been extended to describe molten polymers and concentrated solutions by postulating an ever- changing network in which the junction points are temporary, formed by adjacent strands that move together for a while and then gradually pull apart. It is necessary in the theory to make some empirical statements about the rates of formation and rupturing of the junctions.
2. The single- molecule theories were originally designed for describing the polymer molecules in a very dilute solution, where polymer- polymer interactions are infrequent. The molecule is usually represented by means of some kind of “bead spring” model, a series of small spheres connected by linear or nonlinear springs in such a way as to represent the molecular architecture; the bead spring model is then allowed to move about in the solvent, with the beads experiencing a Stokes’ law drag force by the solvent as well as being buffeted about by Brownian motion. Then from the kinetic theory one obtains the “distribution function” for the orientations of the molecules (modeled as bead spring structures); once this function is known, various macroscopic properties can be calculated.

Polymeric melts

The behavior of polymeric melts are

1. Non- Newtonian polymer melt
2. Linear and non- linear viscoelastic behavior
3. Isothermal and non- isothermal melt behavior

Convective momentum transport

Assumption is a volume element of the polymer melt ∆x∆y∆z of arbitrary shape.

The viscosity arises from a consideration of the random motion of the molecules in the fluid- i.e., random molecular motion with respect to the bulk motion of the fluid. Momentum can, in addition, be transported by the bul flow of the fluid, and this process is called convective transport.

Combined momentum flux tensor Φ, which is sum of the convective momentum flux tensor ρvv and the molecular momentum flux tensor π and that the latter can be written as the sum of pδ + τ, v is the velocity vector of the fluid

Φ = π + ρvv = pδ + τ + ρvv

The equation of continuity

To get the equation of continuity we write a mass balance over the volume element ∆x∆y∆z, which describes the time rate of change of the fluid density ρ at a fixed point in the fluid element. Polymer melt is assumed to be incompressible.

{ rate of increase of mass } = { rate of mass in } – { rate of mass out }

∂ρ = - ( ∇ ∙ ρv )

∂t

∂ρ is the rate of increase of mass per unit volume

∂t

- ( ∇ ∙ ρv ) is the net rate of mass addition per unit volume by convection also called “divergence of ρv, ρv is the mass flux.

For incompressible fluid = ( ∇ ∙ v ) = 0

The equation of motion

To get the equation of motion we write a momentum balance over the volume element ∆x∆y∆z,

{ rate of increase of momentum ] = { rate of momentum in } – { rate of momentum out } + { external force on the fluid }

∂ρv = - [ ∇ ∙ Φ ] + ρg

∂t

Where ρv is the momentum per unit volume at a point in the fluid, ρg is the external force per unit volume at a point in the fluid,

∂ρv is the rate of increase of momentum per unit volume,

∂t

- [ ∇ ∙ ρvv ] is the net rate of momentum addition by convection per unit volume,

- ∇p – [ ˅ ∙ τ ] is the net rate of momentum addition by molecular transport per unit volume

∂ρv = - [ ∇ ∙ ρvv ] -∇p – [ ∇ ∙ τ ] + ρg

∂t

The equation of mechanical energy

The equation of change of kinetic energy:

∂ ( 1/2ρv² ) = - ( ∇ ∙ 1/2ρv²v ) – ( ∇ ∙ pv ) –p ( - ∇ ∙ v ) – ( ∇ ∙ [ Ï„ ∙ v ] ) – ( -Ï„ : ∇v ) + ρ ( v ∙ g )

∂t

-( ∇ ∙ pv ) =net rate of work done by pressure of surroundings on the fluid,

-p(- ∇ ∙ v )= net rate of reversible conversion of kinetic energy into internal energy,

-(∇ ∙ [ τ ∙ v] ) = net rate of work done by viscous forces on the fluid,

- ( - τ : ∇v ) = net rate of irreversible conversion of kinetic energy into internal energy and reversible conversion to elastic energy.

ρ ( v ∙ g) = net rate of work by external force on the fluid,

- ( Ë… ∙ 1/2ρv²v ) = net rate of addition of kinetic energy by conversion per unit volume,

∂ ( 1/2ρv² )= rate of increase of kinetic energy per unit volume

∂t

Ï„ =second-order stress tensor which is symmetric i.e.,Ï„_ij = Ï„_ji

Φ=potential energy per unit mass, defined by g= -∇Φ

∂ (1/2ρv² + ρΦ) = -(∇ ∙ (1/2ρv² + ρΦv) –(∇ ∙ pv) –p(-∇ ∙ v) –(∇ ∙ [Ï„ ∙ v]) –(-Ï„ : ∇v)

∂t

This is an equation of change for kinetic+potential energy and referred to as the equation of change for mechanical energy.

Solids, such as single noncubic crystals, fibrous materials, and laminates, are anisotropic. Heat flux vector q does not point in the same direction as the temperature gradient.

q = - [ κ ∙ ∇T ]

in which κ is a symmetric second- order tensor called the thermal conductivity tensor.

The energy equation

The conservation of energy follows:

{rate of increase of kinetic and internal energy} = {net rate of kinetic and internal energy addition by convective transport} + {net rate of heat addition by molecular transport (conduction)} + {rate of work done on system by molecular mechanisms (i.e., by stresses)} + {rate of work done on system by external forces (e.g., by gravity)}

This may be written in vector notation as

∂ ( 1/2ρv² + ρU ) = - ( ∇ ∙ e ) + ρ ( v ∙ g )

∂t

The combined energy flux vector e as follows:

e = ( 1/2ρv² + ρU ) v + [ Ï€ ∙ v ] + q

where

( 1/2ρv² + ρU ) v =convective energy flux vector

[ π ∙ v ]=molecular work flux vector

q=molecular heat flux vector

e=combined energy flux vector

The equation of energy,

∂ ( 1/2ρv² + ρU ) = - ( ∇ ∙ ( 1/2ρv² + ρU ) v) – ( ∇ ∙ q ) – ( ∇ ∙ pv ) – ( ∇ ∙ [ Ï„ ∙ v ] ) + ρ ( v ∙ g )

∂t

Where

∂ ( 1/2ρv² + ρU ) =rate of increase of energy per unit volume

∂t

( ∇ ∙ ( 1/2ρv² + ρU ) v) =rate of energy addition per unit volume by convective transport

( ∇ ∙ q ) =rate of energy addition per unit volume by heat conduction

( ∇ ∙ pv ) =rate of work done on fluid per unit volume by pressure forces

( ∇ ∙ [ τ ∙ v ] ) =rate of work done on fluid per unit volume by viscoelastic forces

ρ ( v ∙ g ) =rate of work done on fluid per unit volume by external forces