REFERENCE

NBS

PUBLICATIONS

A 111 Ob 031724

NBSIR 85-3130

A Model Describing the Steady-State

Pyrolysis of Bubble-Forming

Polymers in Response to an

Incident Heat Flux

I. S. Wichman

U S. DEPARTMENT OF COMMERCE

National Bureau of Standards

National Engineering Laboratory

Center for Fire Research

Gaithersburg, MD 20899

May 1985

U S. DEPARTMENT OF COMMERCE

UC~

100

• U56

85-3130

1985

ONAL BUREAU OF STANDARDS

NBSIR 85-3130

NATIONAL BUREAU OF STANDARDS

A MODEL DESCRIBING THE STEADY-STATE

PYROLYSIS OF BUBBLE-FORMING

POLYMERS IN RESPONSE TO AN

INCIDENT HEAT FLUX

I. S. Wichman

U S. DEPARTMENT OF COMMERCE

National Bureau of Standards

National Engineering Laboratory

Center for Fire Research

Gaithersburg, MD 20899

May 1985

U.S. DEPARTMENT OF COMMERCE, Malcolm Baldrige, Secretary

NATIONAL BUREAU OF STANDARDS. Emmt AmbUr. Director
.
TABLE OF CONTENTS

Page

LIST OF FIGURES iv

ABSTRACT 1

f

I. INTRODUCTION 3

II. GROWTH RATES AND TRANSLATIONAL VELOCITIES

FOR SINGLE BUBBLES 5

II. 1 The Translational Velocity 6

II. 2 Growth Rate 8

III. THE BUBBLE DISTRIBUTION FUNCTION 10

III. 1 Solution for the Bubble Distribution Function .... 14

IV. CONSERVATION EQUATIONS IN THE MOLTEN POLYMER 15

IV. 1 Conservation of Mass 16

IV. 2 Conservation of Momentum 17

IV. 3 Conservation of Species 18

IV. 4 Conservation of Energy 19

IV. 5 Nondimensional Equations 20

V. THE CASE OF NEARLY-CONSTANT POLYMER MASS FRACTION 24

V. 1 Regression-Rate Formula 29

IV. DISCUSSION AND CONCLUSIONS 32

NOMENCLATURE 35

REFERENCES 39

iii
LIST OF FIGURES

Page

Figure 1, The model configuration 41

Figure 2. The nucleation and growth of bubbles in the polymer melt.

(a) Small clusters of monomer are formed. (b) The monomer clusters

may nucleate small bubbles, which may grow or collapse. (c) The

growing bubbles translate in the direction of the local temperature

gradient; they are at all times surrounded by a layer of monomer. Bubble growth takes place through vaporization at the interface 42

Figure 3. Energy balance at the surface 43

iv
ABSTRACT

A theoretical model is developed to describe the in-depth effect of

bubbles on the steady-state transport of volatile gases (monomer) from the

surface of a polymer subjected to an incident heat flux. In this model the

effect of the bubbles on the surrounding (liquid) polymer is felt through the

bubble number distribution function, n, which appears in the equations for

conservation of mass, momentum, species and energy in the melt. The equation

for the evolution of n includes the effects of bubble growth, convection and

nucleation; its derivation requires preliminary study of the growth and motion

of an individual bubble in a liquid with a temperature gradient. With these

equations, formulas for the mass flux of volatiles from the polymer surface

and the bubble void fraction are developed, for the special case of constant

polymer mass fraction.

- 1 -
'
I. INTRODUCTION

Although the regression and degradation of thermoplastic materials in

response to external heating has been extensively studied [1-4], none of the

previous models consider the influence of the in-depth mass-transfer process

on the steady-state regression rate. In all of these models the heated

condensate is assumed to remain solid until it finally gasifies at the

surface, even though degradation reactions for the thermoplastic are included

in the formulation. Consequently these models describe only the thermal and

chemical mechanisms of polymer gasification.

However, recent experimental study of the gasification of thermoplastics

[5] indicates that the rate of volatile gas evolution from the heated surface

of the regressing polymer depends very strongly on physical processes

occurring inside the heated material. Specifically, polymers such as polyethylene (PE) and polymethylmethacrylate (PMMA) are observed to liquify (with

negligible thermal expansion) in response to an incident external heat flux,

thereby allowing the nucleation of bubbles that grow and transport volatile

gases (monomer) to the surface by moving through the molten layer. These indepth processes allow a substantial increase of the regression rate over that

obtained when gasification occurs merely by surface vaporization.

The model configuration studied here is shown schematically in Fig. 1.

The incident heat flux, Q, through the gas causes the surface of the

infinitely thick non-char-forming polymer to attain the temperature T and to

s regress at a steady rate r. An increase of the heat flux generally increases

both T

g and r. In Fig. 1 the coordinate system is fixed to the regressing

-3 -
surface. The incident heat flux initiates the thermal degradation reaction

P M (polymer + monomer) at some point, x < 6^, in the solid, thereby

enabling the formation of small localized clusters of monomer. At the position x = 6^ (see Fig. 1), the saturation (boiling) temperature of the monomer

is attained (T ~ 100°C for the monomer, MMA, of PMMA [5]). Since a significant quantity of undegraded (or only partially degraded) polymer still exists

at the surface [6], the surface temperature, T

g

, attains values much greater

than the saturation temperature of the liquid monomer, T^.

Therefore, as the surface is approached the local temperature rises well

above the monomer superheat limit and homogeneous nucleation of bubbles in the

monomer clusters in x > 5, occurs spontaneously (note that heterogeneous

D nucleation may occur at sites of chemical impurity, at inclusions, etc.

throughout the heated polymer), see Fig. 2. Consequently the degree of superheating and the bubble nucleation rate both increase while the viscosity and

surface tension of the liquid (the principal obstacles to bubble growth in the

early stages) both decrease. The nucleated bubbles then grow by monomer

vaporization, thermal expansion and collisions with other bubbles, and finally

release their contents to the gas at x = 0.

The practical importance of the configuration shown in Fig. 1 is its

fundamental similarity to that occurring in laboratory-scale flame spread

across thermally-thick polymers. In these problems the heat flux, Q, is

supplied by the exothermic reaction at the gas-phase flame; the fuel for the

flame is in turn supplied by the release of volatiles from the polymer

surface. Clearly, the rate of volatiles release affects the flame spread

-4 -
rate. The actual flame-spread configuration differs from that of Fig. 1

because Q (which normally is a combination of conductive and radiative heat

fluxes to the surface) varies with the downstream distance from the flame

front (since the distance between the flame and the surface changes).

Consequently, the one-dimensional configuration of Fig. 1 is only locally

valid.

In the following section expressions for the growth rate and translational velocity of an individual bubble in a variable-property viscoelastic

medium are derived. These expressions for an individual bubble are then used

in Section III to derive an equation for the evolution of the bubble number

distribution function, which describes the influences of convection and bubble

nucleation on the changes in the bubble distribution in the molten polymer.

Section IV then discusses the influence of the bubble distribution on the

flu id -dynamical behavior of the polymer melt (the bubble distribution function

enters the conservation equations for the molten polymer through the source

and sink terms contained therein; for example, the overall effect of the

bubbles on the equation for conservation of mass is to produce a mass sink ).

An analytical solution for the special case of constant polymer mass fraction

is then presented in Section V.

II. GROWTH RATES AND TRANSLATIONAL VELOCITIES FOR SINGLE RUBBLES

Before proceeding to a detailed discussion of the effects of the bubble

distribution on the regression rate, the behavior of an individual bubble In a

viscoelastic medium must be satisfactorily described. A spherical bubble

generally has two components of motion, translational and radial. Expressions

-5 -
for the translational velocity v and the radial growth rate ft are obtained

fSJ herein from the extensive review literature concerned with the flow of fluid

particles through viscous media [7-9] and with bubble dynamics [10-14],

respectively. In the literature these two components of motion are examined

independently; thus, analyses of fluid-particle flow through viscous media

focus on the flu id -mechanical description of the flow fields Inside and around

fluid particles of fixed shape, while the literature of bubble dynamics deals

only with bubble growth, usually in spherically-symmetric flow fields. Some

modifications of these previous calculations will clearly be necessary here,

since the bubble translational motion and growth rate, in a liquid with a

temperature gradient, are actually coupled.

II. 1 The Translational Velocity

It has been experimentally observed that small bubbles move In the

direction of the local liquid temperature gradient. This motion occurs

because of the stresses induced by the gradient of the surface tension. A

local increase in the surface temperature produces a local decrease of the

surface free energy (the surface tension), a, which generates a non-uniform

tangential stress in the surface, in a direction opposite to the temperature

gradient. Therefore, the direction of the surface flow in the bubble is away

from the warmer pole and toward the cooler pole, so that the bubble moves in

the direction of its warmer pole with respect to the liquid.

An analytical expression for the translational velocity of the bubble is

obtained by examining the steady-state, low-Reynolds-nuraber limit of the

Navier-Stokes equations. In addition, the local temperature gradient is

-6 -
assumed linear, the local fluid viscosity is assumed constant and the variation of the surface tension is assumed linear in the temperature gradient,

viz. , Va = (da/dT) VT. The condition of axial symmetry can be applied when

the gravitational vector and the temperature gradient are parallel. Thus [9],

v (x, R) ( 2 . 1 )

where x is the direction of motion, g is the gravitational constant, R is the

bubble radius and v = p/p is the kinematic viscosity of the liquid. For small

2 bubbles the buoyancy-induced component of the translational velocity, gR /3v,

may be neglected in comparison with the thermal-gradient component. By nondimensionalizing Eq. (2.1) and then dividing the respective nondimensional

groups, one finds that this condition is satisfied when

( 2 . 2 )

Here E

Q

is the Eotvos number, L

Q

is the characteristic conduction length for

large changes in the temperature gradient and D

Q = 2Rq is the characteristic

bubble diameter. For small bubbles in molten PMMA, reasonable estimates

3 -2 -1 are p ~ 1 g/cm , R^ ~ 10 cm, ~ 10 cm and a ~ 20 dyne/cm, which gives

o o o o

E

q ~ 10 ^ an(j R

q

/L

o ~ 10 thereby satisfying the conditions of Eq. (2.2).

Therefore, the expression for v used henceforth is

v (x,R)

R

2p(x)

/v (- §) r * (2.3)

Viscosity gradients cannot, by themselves, exert forces on the bubbles.

Therefore the assumption of constant viscosity does not have a significant

effect on the subsequent formulas for v.

-7 -
where the variation of y with position x has been included. Note from Eq.

(2.3) that v increases when R and VT increase and when y decreases; the ' r\J ' I /v 1 bubbles accelerate as the surface is approached.

II. 2 Growth Rate

There are many possible expressions for ft (x, R), depending on whether

the particular growth regime under consideration is dominated by viscous,

inertial or thermal effects [11-14], Bubble growth rates also depend on

additional factors such as the concentration of dissolved gases in the liquid

[15], its viscoelastic behavior [16] (i.e. , whether or not it is Newtonian)

and whether it is a single-component fluid or an emulsion [17]. In this study

it is assumed that the liquid polymer is a single-component Newtonian fluid

containing no dissolved gases. When the conditions of spherical symmetry

apply, the equilibrium radius of a vapor bubble that has nucleated in such a

medium is

R = 2o/[p

v

(T) - p] (2.4)

where pv

is the bubble vapor pressure and p is the ambient liquid pressure

(p = p - p(g • x) for the regressing surface, where p. is the gas pressure at

o ~ ~ o the surface, see Fig. 1). In Eq. (2.4) the surrounding liquid temperature T

must be greater than the bubble vapor temperature T

v

. The quantity AT = T-T^

is called the liquid superheat.

After the bubble has nucleated, the primary obstacle to radial growth in

this initial growth stage is the opposing action of the surface tension. In

-8-
this stage p v >> p, so that Eq. (2.4) gives R ~ 2a/p^; assuming p^ > 10 atm

and a ~ 20 dyne/cm gives R < 0 (10 ^ cm). The next stage of bubble growth is

controlled by liquid inertia. This stage is followed by an intermediate

•Up growth stage in which both inertial and thermal effects dominate ; a

continuous influx of thermal energy is required to provide the additional

vapor for bubble growth. In these stages of growth the surface tension terra

-4

in Eq. (2.4) is 0(1); using p >1 atm gives 2R < 10 cm. Finally thermal

V ^ 'W effects alone control the growth process. In this final stage numerous

authors [18-20] have shown that

ft = /3/ ir [ A/ AT //at , (2.5)

where and a are the latent heat of vaporization and the thermal diffusivity

of the liquid, respectively. Thus, ft is obtained as an approximate balance

2 — between the rate of heat flow into the bubble from the liquid, 4-itR AAT//at ,

and the latent heat required to supply the vapor in the bubble [14], 4irR^ p^ft.

In this final growth stage the surface-tension term in Eq. (2.4) becomes

negligibly small; therefore 2R » 10"^ cm.

Obtaining simple analytical expressions for ft analogous to Eq. (2.5) in

the inertia-controlled and the thermal/inertia-controlled growth stages is

generally not possible because the governing equations of motion are nonlinear.^ But since the average bubble radius in its initial growth stages,

Rj, is at least one to two orders of magnitude smaller than in the final

*Diffusional effects need not be considered because the bubble is assumed

always to be surrounded by a thin layer of liquid monomer (see Fig. 2).

simple interpolation formula for R(t) has been suggested [21],

However, the physical grounds for deriving this formula are weak.

-9-
(thermal) growth stage, R^, (l.e. , R_/R_ < 10 *), its overall effect on the

transport of mass and momentum in the mixture is smaller by a factor of at

least 10 for mass transport (since p^V <* R ) and at least 10~ for momentum

4

transport (since p^Vv <* R ). Consequently the detailed modeling of bubble

growth in the small-R regimes appears unnecessary, and Eq. (2.5) is believed

sufficient.

The integration of Eq. (2.5) with respect to t gives R <= t*'^; thus

K(x)

R (x, R) = — , (2.6)

2 2 2 where K(x) = (

/tt

)

a (p/pv

) (c^AT/i^) = (6/ir) aA is now considered a function

of position because of the changes encountered in p, p^, c^. A, AT and «£ as

the bubble translates through the liquid. Here A = (p/pv )(c^ATAC) is the

Jakob number. Note that the effects of property changes on &, arising from

the bubble motion, can also be accounted for by replacing K(x) by K [ 1 + e(x)],

where K is now constant and e, the correction factor, must be obtained empirically. By analogy with droplet flows in sprays [22], e ~ Re^Vr

11

, where

Re = 2vR/ v is the Reynolds number and Pr = v/a is the Prandtl number. Other

forms for e are, of course, also possible.

III. THE BUBBLE DISTRIBUTION FUNCTION

The calculations of the preceding section are now extended to derive,

physically, a statistical equation for the evolution of the bubble number

distribution function; this equation will subsequently be used to describe the

overall influence of the bubbles on the pyrolysis process. To provide such a

- 10 -
statistical description it is necessary to specify the number distribution

function, n(x, V, t )dxdV, which gives the probable number of bubbles in the

* 'V volume range dV of V and in the coordinate range dx of x at time t. Before

deriving the evolution equation for n, however, various physical assumptions

relevant to the configuration of Fig. 1 must be stated and discussed.

3

It is assumed here that the bubbles are spherical [V = (4/3 )tt

], that

the surface-regression process is steady-state and one-dimensional (i.e.,

3

( • ) / 3 1 = 0 and V • (•) = 3(0/8x etc.), that the effects of bubble breakup

are negligible and that the rate of binary bubble collisions is negligible.

These assumptions are now discussed in turn.

The bubbles will be spherical when the Reynolds number for the flow

around each rising bubble, Re = 2R v /v , is small compared with unity, oo o regardless of the values of the Eotv’os number, [Eq. (2.2)], and the Morton

4 3 number, M = gy^/p^a^ [7]. For PMMA reasonable values of v

q

, R

Q and vQ are

2 -1 -2 -4

v ~ 10 cm /s, 2R^ <10 cm and v ~ 10 cm/s, whereby Re < 10 ; therefore

o

’ o ~ o ~

these bubbles will always be spherical.

Surface regression occurs as a steady-state process when the thickness,

t, of the condensed-phase sample is much greater than the thermal wave penetration depth, L

g = X/pc^r. The steady-state regression of PMMA and PE

samples can only be achieved for very large heating times, t - ® [5], when the

incident heat flux is radiative. This occurs because these polymers are very

nearly optically transparent to the short-wavelength near-infrared radiation

Note that since the bubble velocity, v, depends on x and V [Eq. (2.3)], its

components cannot be considered as additional independent variables in n.

- 11 -
used to heat the surface in these experiments; the radiant preheating

continually increases the thermal wave penetration depth (and therefore the

regression rate, r) until the final (asymptotic) value is attained [5].

The assumption of one-dimensionality corresponds physically to the

neglect of temperature variations in the transverse (y-z) plane compared with

variations in the vertical coordinate, x [see Fig. 1], Therefore the sides of

the sample must be insulated and the temperature perturbations caused by

neighboring bubbles must be negligible. In this case probabilistic variations

in the dependent variables (such as the bubble number distribution function,

the liquid density, pressure, velocity, enthalpy, etc...) and in the thermophysical and transport properties such as specific heats and thermal

conductivities occur only in x.

For a stagnant liquid in which the large-scale effects of turbulence and

velocity gradients are negligible the only remaining mechanism for bubble

breakup is the growth of a Rayleigh-Taylor instability in the bubble surface

[7], For a translating bubble the leading surface becomes unstable when the

wavelength of a disturbance there exceeds the critical value, X . /p g. crit o

K o The approximate criterion for stable disturbances is E < 16. For FMMA

o ~

3

p ~ 1 g/cm and a ~ 20 dyne/cm, whereby R < 0. 3 cm for stability. This

o o o ~

condition will always be satisfied in this study.

When the bubble number density is high and when many disparate bubble

sizes are present the effect of collisions cannot be neglected. Under the

conditions studied here one can write, by analogy with the theory of falling

droplets [23], =: ir (R^ + R

2

) [v^ - V

2

] for the bubble collision frequency

-12-
factor, where and respectively, represent the radius and velocity of

bubble i. Note that Rj > R

2 is required here, since v^ ~ R^ and > 0.

Therefore when Rj - R

2 << 0(1) the collision term becomes negligible. An

alternative criterion for neglecting collisions is R

Q /L « 0(1), where L is

—1/3

the average distance between bubbles. Defining L ~ (N ) , where N

Q is a -2

representative value of the bubble number density, and putting R ~ 10 cm

3 3 -1

and N ~ 10 bubbles/cm gives R /L ~ 10 « 0(1); thus the bubble distribuo o tion must be disperse.

Under these assumptions the equation for the evolution of the bubble

distribution function becomes

9(nft) 3(nv)

= Q

3R 3x (3.1)

where the terms 9(n&)/9R and 3(nv)/3x represent the flux of n through R-space

and physical space, respectively. Note that n is now considered to be a

function of x and R. Note also that Eq. (3.1) can be derived by taking the

derivative, with respect to R, of the more-general integral equation

(nft)R

_3_

3x f n(x,s)v(x,s)ds = (nft)

R*

R* ! (3.2)

where the term on the right-hand side represents the nucleation rate. Here,

R* is the critical cluster size for nucleation, which is two to three orders

of magnitude smaller than the smallest macroscopic bubble shown in Fig. 1.

Since Eq. (3.2), as written, is quite general, any of the numerous forms

for (n&)R£ derived in the extensive review literature of nucleation theory can

be used [24-26]. In this study, however, a simple expression (obtained from

the so-called classical theory of nucleation) is employed.

- 13-
In this theory the nucleation process is modeled as a kinetic mechanism.

n tv which takes place through the stepwise agglomeration process, + A^ +• * where A^ represents an activated cluster of j single monomer molecules and Aj

is an unactivated monomer molecule. Therefore, (nft)R* is equal to the product

2

of the equilibrium concentration of critical clusters, F • exp [-4ttpR* /3kT],

with the characteristic pre-exponential rate constant, /2<j/Trm, viz. [27,28],

,2

(nR)

R* = F • /2a/iTm e

4irqR*

3kT

(3.3)

= C(x)e-H(x)/T(x)

Here, F is the number of liquid molecules per unit volume of the original

liquid, a is the surface tension, k is the Boltzmann constant and m is the

average molecular mass. The pre-exponential factor, C(x), becomes vanishingly

small for large, negative x (see Fig. 1), because F -» 0 there. Similarly, the

exponential factor becomes small when R* » /3kT/4ira ; for PMMA this condition

is satisfied (using a ~ 20 dynes/cm, k ~ 1.4 x 10 ^ erg/°K and T ~ 10^ °K)

when R* » 10-

^ cm.

III. 1 Solution for the Bubble Distribution Function

Equation (3.1) can be solved, with the aid of Eqs. (2.3), (2.6) and

(3.3), to give

n(x,R) _R J(z ) K(z) J(x) C(z) -H(z)/T(z)

e , (3.4)

where the variable z is defined by the equation

- 14 -
(3.5)

Note that the condition C(-®) * 0 gives lim n(x,R) « 0 and lim n(x,R) = 0.

When K/J in Eq. (3.5) is constant, then the variable z takes the simple

form

obtained from Eq. (3.5) by taking the limit R*/R - 0. In spite of this

simplification, however, the effect of the bubble number distribution on the

conservation of mass, momentum, species and energy in the polymer melt [which

enters Eqs. (4.3), (4.7), (4.11) and (4.12) through integrals over R] is still

very complicated, since n contains the dependent variable T(z) in the

exponent.

IV. CONSERVATION EQUATIONS IN THE MOLTEN POLYMER

Equation (3.4) is coupled to the conservation equations for the molten

polymer through the various sipk terms, which describe, in an average sense,

the effect of the bubble distribution on the loss of mass, momentum, species

and energy from the liquid to the bubbles. In what follows it is shown that

the usual equations of fluid dynamics describe the changes in the average

liquid properties, provided that the proper source terms are added to account

•flf _ for the average effect of the bubbles. The precise form of the source terms

A rigorous evaluation of the effect of the source terms on the liquid

properties would involve application of the boundary conditions at each

bubble and then the computation of an ensemble average [22]. This procedure

is, for this study, unnecessarily complex.

R-xo

. fvJ \ p

3 z = x "

(3 k) r (3.6)

- 15-
used here can be deduced from the equations for conservation of mass, momentum

and energy in the vapor-phase component of the two-phase mixture. These

equations are obtained by multiplying Eq. (3.1) by p^V, p^Vv and

2

PvV(h

v + v /2), the mass, momentum and energy, respectively, of a single vapor

3 bubble of volume V - (4/3 )ttR and density p^, and then integrating over R.

Although the details of this calculation are not pursued here, it is straightforward to show that the source terms arising in the vapor-phase equations are

identical to the sink terms appearing in the corresponding conservation

equations for the liquid [on the right-hand sides of Eqs. (4.3), (4.7) and

(4.12)]. Instead, the derivation presented here is phenomenological.

IV. 1 Conservation of Mass

The loss of mass from the liquid to a bubble of density p v (x)^ and

3 volume V = (4/3)ttR is - d (pvV)/dt; therefore for all bubbles the mass loss

from the liquid is

oo d(p V) - f n — — dR, (4.1)

o dt

where n = n(x,R) and

d( »)

dt

= R v 3(0

3x

* (4.2)

The vapor density pv

is given by pv

= PV/& T

V

> where Tv

= T

v

(x) [13]. From

Eq. (2.4), p v

=* p since the surface-tension term is negligibly small. But

p = pQ - p (g • x), where pQ is the pressure in the gas at x=0. Thus

pv

= pv

(x). Note also that dp

v /dx * -pvd(lnT

v

)/dx.

- 16-
The equation for conservation of mass for the liquid is then

d(p

f

u)

dx

*(p

vv) dt

dR, (A. 3)

where u is the velocity of the liquid and p^, the fluid density, is the mass

of liquid per unit volume of physical space. Since the volume fraction of

00 physical space occupied by the bubbles is / n(x,R)VdR, the fluid density and

0

the liquid density [the mass of liquid per unit volume of liquid] are related

by

1 ~ P- f n(x,R)R3dR = 1 - H.(x),

0

(A. 4)

where

II (x) = f n(x,R) Rm

dR

ra 0

(A. 5)

is the m

c moment of the bubble distribution function. By using Eqs. (2.3),

(2.6), (A. 2) and (A. 5), Eq. (A. 3) reduces to

k (pf“) - -

Ait dpv

Airp Kn, + j— Jn, v 1 3 dx A

(A. 6)

IV. 2 Conservation of Momentum

The equation for conservation of momentum is

d f 2 x d? ( p

fu + P + Txx^>

oo nd(p Vv) 00 - /

~r~ dR = - f nFdR,

0 0

(A. 7)

- 17 -
where the shear stress, t , for a Newtonian liquid with negligible bulk

XX

viscosity, is given by

TXX

4 du

3

p dx ’ (4.8)

and F = d(p

vVv)/dt represents the net force per unit mass exerted on the

bubble by the fluid. In general, F can be decomposed into the components Fp

and Fr , opposing bubble translation (i.e. , F._ = p Vdv/dt is the drag force on

the bubble) and bubble growth (F^ = vd^p^V^/dt). The drag force, F^, on a

bubble of radius R and velocity v(x,R) is given by [7]

F

D (x,R) = - 4irp(x) Rv(x, R). (4.9)

Therefore with the assistance of Eqs. (2.3), (2.6), (4.2), (4.5), (4.8) and

(4.9), Eq. (4.7) becomes

d_

dx 4TrpJll2 J 4up Kn 0 v 2

4tt

** pv

3 dx

IV. 3 Conservation of Species

(4.10)

With Eqs. (4.3) and (4.6) as guides, one writes

d(p V)

dx t p f

( U + W= W

k

" \ I n ~~ dR » k = 1 M

= w. - Q. k k

j^47rp vKn

1 k - l, ..., m (4.ii)

- 18-
for the conservation equations of the M polymer species. Here Y

k- uk> wk

and are the mass fraction, the diffusion velocity, the rate of mass creation by chemical reaction, per unit volume, and the mass flux fraction across

the bubble interface of species k, respectively. Note that E w, = 0 and

k

E = 1 ; both the w^ and the are generally functions of position, x. k

IV. A Conservation of Energy

The loss of energy from the liquid to an individual bubble is accounted

for by the sink term -d(p

vVhv + p vVv^/2)/dt, where h^ = h^^ + C

p V

(T “ T^) is

the bubble vapor enthalpy. Therefore the equation for conservation of energy

for the liquid becomes

d_

dx [pfu

(

h + f") + q + UTxx]

= " f n ^[pv^v + P V

V i“] dR (A. 12)

( 0 ) where h = n + c (T - T^) = E h^Y^ is the liquid enthalpy and

q = - XdT/dx + p^ E h^U^Y^ is the heat flux. With these definitions of h and

k

q and Eq. (A. 8) for txx

» Eqs. (A. 3), (A. 11) and (A. 12) can be combined to

produce an energy balance that includes the chemical reaction term, E h^w^.

k

Then, noting that the enthalpy of the liquid, evaluated at the bubble interface, is given by E h^ft^ = h , where hm

is the monomer enthalpy (see the

k *

previous footnote; = 0 for all k ^ m), and that h

v

(x) - hm(x) =*^

m

(x),

where

m

(x) > 0 is the latent heat of vaporization of the liquid monomer, and

using Eqs. (2.3), (2.6), (A. 2), (A. 5) and (A. 9), the liquid-phase equation for

conservation of energy becomes

The bubble interface is defined as a region in which the fluid properties

differ appreciably from their local average values. This definition applies

especially for the average local molecular weight W of the liquid; near the

bubble W is assumed very nearly equal to (but still greater than) the monomer

molecular weight.

- 19 -
[

i-C u

f p

C pk

Y

kUkj dx

d_

dx (£)-*(>£)- h, w. k k

^uJ2n

3 -(j£m

- + y

1 yy Jn

4

j

r [

4 ’", vKn

3 + r- d^T Jn

6j (

cpv It)t 1Tpv

J^4' (4.13)

The first three terms on the right-hand side are familiar, and describe the

effects of thermal diffusion, chemical reaction and the viscous-stress work,

respectively. The fourth terra represents the addition of energy to the liquid

through bubble friction, while the last term describes the convective bubbleenergy flux. The sum of the terms containing u and J represents the net

kinetic energy lost from the liquid for bubble growth: when d(p

vV)/dt = 0

these terms vanish. Finally, the terra containing [note that the factor in

square brackets is simply the mass-loss term of Eqs. (4.6) and (4.10)] represents the net vaporization energy lost from the liquid for bubble growth.

IV. 5 Nondimensional Equations

To assess the importance of the various terms in Eqs. (4.6), (4.10),

(4.11) and (4.13) they are nondimensionalized by using W = u/r, M. = n/nQ

,

I

= a/a

g

,/2= R/R0 , K = x/L

g

, * = p

f /pg

, <f> v= pv

/pvs

» n = u/p

s

> 9 = T /T

s

»

X - VcpS> h/ VT

s- x - X/X

s

, < = K/K

s (i.e., fc o = Ks /R

o

), J/J

s (l.e., v

Q - JS

R

C

), l m

/c

ps

T

s

, - a>k/(ps r/L

s

), = p/p

s r

2 and

it = II /n R

m+^. Here r is the regression rate, R^ is the average value of R m m o o

’ o in the thermal growth stage, L = a /r is the characteristic liquid conduction

® s length and a , p , p , y , T , c and X are, respectively, the characters s vs s s ps s istic values of the surface tension, the liquid density (mass of liquid per

-20-
unit volume of liquid), the bubble-vapor density, the dynamic viscosity, the

liquid temperature, the specific heat and the thermal conductivity, evaluated

at the surface. Comparison to Eqs. (2.3) and (2.6) shows that J = a /2y L

S S S 8 2 and K * (6/ir)a A , where A is the value of the Jakob number evaluated at the

s s s s surface.

The nondimensionalization of Eqs. (4.6), (4.8), (4.11), (4.13) and (3.1)

according to this prescription produces the following nondimensional groups as

parameters in the equations:

T, = 3 (p /p )(L /R /r)(n R )V , 1

' Kvs

Ks

;v- s o''- o o o' o (4.14)

r

2 - (1/3) (R

o /Ls )(vo

/fto

), (4.15)

T, = 3 (v /r)(L /R rfn R )V

3 o s o' K o o' c (4.16)

r 4 = V

o /r » (4.17)

T- = r /c T , 5 ps s (4.18)

and

Re = L r/v

s s (4.19)

Here represents the characteristic magnitude of the mass loss to the

bubbles for radial growth [see, e.g. , the first term on the right-hand side of

Eq. (4.6)] while ^

is the ratio of the characteristic magnitudes of the

-21 -
second and first terms on the right-hand side of Eq. (4.6); it may be anticipated that « 0(1), since the translational velocity is generally smaller

than the bubble growth rate. The parameter represents the ratio of the

drag-force term [the first term on the rhs of Eq. (4.10)] and the shear-force

terra [the last term on the lhs]. The ratio of the characteristic bubble

velocity and the regression rate is given by r^, which appears in the momentum

and energy equations. The quantity r^, which appears in the energy equation

[Eq. (4.13)], is the ratio of the kinetic energy of regression to the available thermal energy near the surface, and is generally very small. The

parameter Re is the Reynolds number, and multiplies the convection and

pressure-gradient terms in Eq. (4.10). Since L = a / r, the Reynolds number

s s is the inverse of the Prandtl number.

Reasonable physical estimates for the parameters defined by Eqs. (4.14)

_3

to (4.19) are obtained for PMMA by putting X ~ 10 cal/cm-s-°K [29],

s

p ~ 1 g/ cm\ c ~ 1 cal/g-°K, T ~ 5 x 10 2 °K, n Rrt ~ 10 3 bubbles/cm3

, S po s o o V

Q ~ 10 5 cm\ r ~ 10"2 cm/s [5], p^g ~ 10

3 g/cm3

, v

g ~ 10 cm

2 /s and

a ~ 20 dyne/cm. Therefore, the characteristic conduction length is

S L = a /r ~ 1 cm, the characteristic bubble translational velocity is

s s vososo = JR = aR/(2yL')~10 cm/s and the characteristic bubble growth rate v ss ;

is &oso = K /R = (6/ir)a A /R ~ 10 A cm/s; assuming A < 0(1) [which, from sso s s~

the integrated version of Eq. (2.5), indicates that in t ~ 10 seconds the

bubble grows from R = 0 to R ~ 10

-2

cm] gives R < 10 * cm/s. Consequently,

o ~

r

j ~ o(io

_1

), r

2 > o(io~3

), r

3 > o(io 2

), r 4 ~ o(io), r

5 ~ o(io“9

) and

r -4>

Re ~ 0[10 J. With these estimates the equations for conservation of mass,

momentum, species and energy in the melt reduce to

-22-
Uu - A, (4.20)

)

d_

d£

d

d£ L"k

Aj 1, ..., M

and

o[U+ * \\K ] If " (x If)

- Uy\ - *».

respectively , where

A(£) = T

x

[<(.v

Cc)ic(5)-ir

1 (c)]

and

(4.21)

(4.22)

(4.23)

(4.24)

r(5) H fr 3 [n(5)^ (5)w

2 (5)]- (4.25)

The terms containing A in Eqs. (4.20) and (4.22) and r in Eq. (4.21)

represent, respectively, the effect of the mass loss from the liquid for

radial bubble growth on mass and species conservation, and the effect of the

overall bubble drag force on momentum conservation. The term containing A in

Eq. (4.23) represents the loss of vaporization enthalpy from the melt to the

bubbles, by virtue of the mass flux across the interface.

The nondimens ional form of Eq. (3.1), for the bubble distribution function n is

aQ*A) aQttv) = n

3/1

l 2 dK

-23-
V. THE CASE OF NEARLY-CONSTANT POLYMER MASS FRACTION

In this case a solution for the regression rate is obtained from Eqs.

(4.20) to (4.23) by assuming that the mass fractions of the heavy components

in the molten liquid remain very nearly constant during pyrolysis. Therefore

the mass fractions of the lighter components of degradation must always be

very small. Under this assumption the individual clusters of monomer that are

formed during degradation must immediately nucleate monomer bubbles, since the

nucleated bubbles and their interfaces do not belong to the liquid. As a

result, the mass fractions of the light components in the liquid can be very

small even though the degradation reactions are proceeding rapidly. This

assumption is supported by recent measurements of molecular weight distributions in PMMA samples subjected to an incident radiative heat flux [6].

Therefore, the one-step irreversible reaction P - M (polymer monomer)

is considered, for which the reaction term [see, e. g. , Eqs. (4.11) and (4.13)]

is given by w

p = “ w

m

= “ AYTM exP [ “E Hk. T ] , where the activation energy, E, is

much larger than the molar enthalpy, T, i.e. , E/&T » 0(1). The polymer

mass flux into the bubbles, S2 p

, is negligible compared with the monomer flux

(see Fig. 2), so that SI = 1 - SI “ 1. In addition, the diffusion velocities m p

in the liquid polymer are assumed negligible and the thermal conductivity of

the liquid is assumed constant. In what follows it is useful to introduce the

temperature, 0, as the new independent variable. With this definition the

inner chemical-reaction/bubble-formation zone is defined by the closeness

of 0 to the surface temperature (0 = 1), and not by the physical proximity of

the reaction zone to the surface, £ = 0. Thus, putting p = d0/d£ into Eqs.

(4.20)

, (4.21), (4.22) (for the polymer only, since Ym

+ Y

p = 1) and (4.23),

-24 -
and using d/d£ = pd/d0 (in addition to the previous assumptions), one obtains

mass

d(<t>u)

de

“ A, (5.1)

momentum:

de ^

pn de

J~

r » (5.2)

species

- -<0Ym

e

S(1 - 1/9)

de p

; p

(5.3)

and

energy: <t>UP = P # - 4£4)Ym

e

6(1 - 1/9) - U, de

(5.4)

respectively. Here *5 = (X

s /ps cpg

r

2

)(A/p

g

)exp[-E/tt.T

g

] is the Damktthler

number, the ratio of the characteristic convection (residence) time.

t =X/pc r,to the characteristic chemical reaction time,

c s

Ks ps

t r = [ (A/p

c )exp [-E/(X, ] ] *, g = E/d£.T

g

is the nondimenslonal activation

energy and = K. -h is the enthalpy difference for the (endothermic)

m p

degradation reaction P - M. The boundary conditions for Eqs. (5.1)-(5.4) are

i) p

= 0, <(> = 1, 'W = 1 , Y =1 when 0*9, \ P 0°

J (5.5)

ii) p=p,Y =Y when 0=1, \ s p ps

where p and Y are surface values. Note that the bubble number distribution o pS

function, n, enters Eqs. (5.1)-(5.4) through the inhomogeneous terms,

A and f.

- 25-
Expressions for the regression rate, r, the fluid velocity, , and the

volume fraction, <j> = p /p , are now derived from Eqs. (5.1 )— ( 5.5) by using the

i s method of matched asymptotic expansions. The largeness of the nondimensional

activation energy, g, is utilized to develop expressions for p, <f> , and Y

p

in

an inner chemical-reaction/bubble-formation zone, where 1 - 9 « 0(1), and in

an outer heat-up zone, where g(l-0) «> and the rates of chemical reaction and

bubble formation are negligible.

In the outer zone A -* 0, exp[g(l-l/9) ] 0 and r ^ 0. The appropriate

lowest-order solutions of Eqs. (5.1) and (5.3) are <J> = 1 , M = 1 and Y

p = 1.

The energy equation becomes dp/d9 = 1, to which the solution is p = 0 - 0 . 00

In the inner zone the variables Y =l-T/g+... and 0=1- 0/g + ... P are introduced, consistent with the assumption that Y

p

in the melt is very

close to unity and that the temperature is close to its surface value.

Therefore in the lowest order Eqs. (5.1)-(5.4) reduce to

mass

d(({> *U ) A. P d0 g

*

momentum:

species (polymer): dU^ ) _ G§ -0

P d0 8 6

(5.6)

(5.7)

(5.8)

and

energy:

dp = _ . $ S ~Q _ A

do A/t

s 8 8

’

' (5.9)

-26-
each of which is now addressed in turn

Comparison of Eqs. (5.6) and (5.8) indicates that

A =<#e“9

. (5.10)

Thus, all of the monomer produced by the chemical reaction is subsequently

lost to the bubbles. Note that the elimination of A from the conservation

equations eliminates, from all of Eqs. (5.6)-(5.9), except Eq. (5.7), the

explicit influence of the bubble distribution function, n.

Since the shear stress in the melt must vanish at the cold edge of the

reaction layer, two integrations of Eq. (5.7) give

0

'll = 1 - /

00 t ?

X

, x f

X

-ffi- " ds. (5.11) p(x)p(x) i e 2

P (s)

When the bubble

2 distribution is highly disperse the quantity r/B > appearing

in the integral, is very small compared with unity. Thus, in the lowest

approximation U - 1, so that the bubbles produce only a negligibly small

deviation of the fluid velocity from the regression rate. This appears to be

consistent with

leads to /U= 1

the two-phase-flow mass balance [30], p

g r = p

g u + P v

v » which

" (Pv

v )/(p s r ) = 1 + 0(10"3

v/r).

With the assistance of Eq. (5.10) and the above result, /U= 1, the linear

combination of Eqs. (5.8) and (5.9), integrated once, produces

P - C

1 - ej + (^vs - i

ps

)a - ). (5.12)

-27 -
Here, the constant of integration has been determined by matching to the

solution in the heat-up zone [which requires lim p(*- nner ) = i

m p(°uter ) _

(inner) O^00 0+1

and lim = 1], while the overall heat-release factor,

A ^ + 1 = [A 1 + [A - k ) -A -A , is simply the differs s

v m p

; s v m; s vs ps ence between the bubble-vapor enthalpy and the undegraded polymer enthalpy,

evaluated at the surface temperature. Note that the nondimensional temperature gradient, p, increases as the bubble volume fraction, 1 - <J> , increases.

00

The solution for p in the inner zone is obtained by integrating Eq. (5.9)

[after substituting Eq. (5.10) for A]. The integration constant is determined

by matching to the solution in the heat-up zone, as before; thus,

) V 1 + 6e

0

, (5.13)

where

6 = (5.14)

The composite expansion for p, which is valid to lowest order in both zones,

is therefore given by

p = (1 - 9 ) 1 + 5e

6(1 6) - (1 - 0). (5.15)

The solution for the bubble volume fraction, 1 - <j>, is obtained by

substituting Eq. (5.13) into Eq. (5.12), viz.,

1 '

(i - O * = (/vs -A „.)

V1 + 6e

8(1 - 1 (5.16)

-28-
Note that since 0 < 0 < 1, the bubble volume fraction is always greater than

OO — —

or equal to zero.

V. 1 Regression-Rate Formula

An expression for the regression rate can be obtained by evaluating Eq.

(5.13) at the surface. Thus, putting p p_, 0 0, and recalling the definition of gives, after rearrangement,

Note that all of the chemical parameters appear under the square-root symbol.

The factor in square brackets represents, essentially, the nondimensional

ratio of the net enthalpy flux into an individual bubble to the net thermal

energy applied to the melt layer; a proper nondimensionalization should make

this factor an 0(1) quantity.

To complete Eq. (5.17) an expression is required for the surface

temperature gradient in the melt, p g. It is not possible to use Eq. (5.12)

for this purpose, because it only applies inside the melt, at least one

characteristic bubble radius, R

Q

, from the surface. Instead, an energy

balance across the interface is considered (see Fig. 3). Assuming negligible

in-depth heat absorbtion, the net incident heat flux, Q, must equal the sum of

the net heat conducted into the bubble/melt mixture plus the net energy flux

carried back into the gas by surface vaporization and bubble-vapor transport,

viz.

,

r (5.17)

-29-
Q » Q + m9/ (5.18)

s s

The quantity can be expressed as the sum of contributions from the

bubbles and from surface vaporization, viz., = + ^ere *

is the bubble mass flux and m^ = m - m^ is the vaporization mass flux. Since

\ " / "(O.R) PvsV gV dR

(ps 0 (5.19)

then

m (<= s r ) A* (^ h - £ ) vs s

; fcp) R ) V TT a ( 0 )

» O

j O 4

(5.20)

The net conductive heat flux into the mixture is

dT

dT

% - Vs fa s +

C

1 - O » s' vs dx s’ (5.21)

where <J» is the liquid volume fraction and the second term on the right-hand

side represents the heat conducted through the vapor phase. By substituting

Eqs. (5.20) and (5.21) into Eq. (5.18), then assuming that (dT/dx)

g =

(dT /dx)_, and then solving for (dT/dx)_ one finds V o S

-30-
Equation (5.22) shows that the effect of the bubbles on the temperature

gradient in the liquid is the result of two competing processes. The first

process, described by the last term in the numerator, decreases X (dT/dx) , s s since an increase in the energy carried into the gas by the bubble vapor

9 efflux occurs at the expense of heat conduction into the melt. The second

process, described by the denominator of Eq. (5.22), increases X (dT/dx) by

s s decreasing the amount of available liquid in which to conduct heat deeper into

the melt. An estimate for the relative importance of these two effects is

obtained by forming the nondimensional quotient (l - $ )/(m /p r) =

(p /p

v )( r/v ^^(OVu^CO)], which, for the conditions studied here, is

0(10 ). Thus, the primary effect of the bubbles is to increase the

temperature gradient in the melt. Note that an increase in the incident heat

flux, Q, also increases the temperature gradient.

By nondimensionalizing Eq. (5.22) and using X /X ~ 0(10 ^), as well as VS s the previous estimates for the two competing processes, one finds

0/ f^SPSS-'S p c rT ) - l ,coo\

p

= 7

. (5.23)

S (t)

Comparison with experiments [5] suggests that the quantity Q/rT„ decreases

with increasing Q. Since also decreases it appears reasonable to assume

p

g ~ constant for the range of parameters considered here. Therefore, the

quantity in square brackets in Eq. (5.17) increases as the incident heat flux,

the void fraction and the surface temperature all increase. The magnitude of

this increase, however, is small in comparison with that of the Arrhenius

reaction-rate factor [see Eq. (5.17)].

-31 -
VI. DISCUSSION AND CONCLUSIONS

The only other quantitative steady-state regression-rate calculations

that can be compared with the calculations of this study are those of LengellG

[1]. Direct comparisons, even here, are difficult, because the underlying

physical processes modeled in the two studies are completely different. In

[1], the regression rate is obtained as a balance between the surface heating

rate, the rate of heat removal to the interior of the solid, and the rate at

which the polymer degrades to monomer through the one-step reaction P -» M.

This reaction proceeds fastest at the surface, where the temperature is

highest. Consequently, the polymer mass fraction there must be very small

(Y « 1), while the monomer surface mass fraction is very close to unity.

ps

However, according to the discussion in the Introduction, such a formulation

is inconsistent with thermodynamic considerations: since the monomer superheat limit is only 100 °C, surface temperatures significantly greater than this

cannot possibly be achieved without spontaneous phase changes [solid -> liquid

-> vapor], due to homogeneous nucleation processes and the associated boiling

phenomena. Thus, the original unheated solid polymer must melt or liquify in

such a way that the superheat limit of the degradation products (monomer) can

be exceeded. In contrast to Lengelld's calculations, the calculations of this

study suggest that the regression rate can be obtained as a balance between

the surface heating rate, the rate of heat removal into the interior of the

condensed phase, and the rate at which monomer is lost from the polymer for

bubble formation. The remaining liquid, most of which is original polymer,

can then be heated to temperatures much higher than the superheat limit of the

liquid monomer. This result appears to be consistent with thermodynamic

considerations and experimental observations [5].

-32 -
In the notation of this study the regression-rate formula derived in [1],

for PMMA, is given by

[-(A A + 1-6 )lnY - hi (l-Y )]" 1/2 \ — L ^ s oo ps s^ ps * f p c p

-EA^T

s e(E/«.T 1 ( 6 . 1 )

Note that the factor appearing under the square-root symbol is identical to

that in Eq. (5.17), and that the polymer mass fraction must be assumed very

small (in contrast to the assumption Y

p ~ 1 made in Section V). Forming the

quotient of Eqs. (5.17) and (6.1) gives

2{k " k + 1 - 0 ) InY - hk (1 - Y ^ rv )] VS ps ; L l S oo J £S s

^ ps J J 1/2

pi - C

1 - o

2

( 6 . 2 )

Using T

g ~ 600°K, ~ 300°K, (hm

- h

p

) g ~ 130 cal/g [31], (hy

- h

p

) g ~ 230

cal/ g, cpg ~ 1/2 cal/ g-°K [29], Y

pg

= 0.01 [1] and p g ~ 1 gives r/rL ~ 3,

indicating that regression rates predicted by Eq. (5.17) are generally larger

than those predicted by Eq. (6.1).

Perhaps more important than comparisons of regression-rate formulas is

the result expressed by Eq. (5.16) for the void fraction, l-$. In the

customary two-phase-flow analyses [30], the void fraction is treated simply as

an additional independent variable, unrelated to the other dependent variables

(such as the temperature). By contrast, Eq. (5.16) defines l-<}> in terms of

the temperature, 9, and the remaining chemical and thermochemical parameters,

9 » k - k , 6 and 0. Equation (5.16) can be used to calculate average OO Vb pb

properties of the two-phase mixture as functions of the temperature and the

thermo chemical parameters, e.g., for any property P, P avg = <t> P

^ lg +(l-$ )P

vap

«

-33-
It is not possible to make direct comparisons between the theoretical

predictions of this study and the experimental measurements of previous

researchers [5], since the theory has been developed under the steady-state

assumption while the experiments were conducted under transient conditions.

As discussed in Sec. Ill, a steady state can be attained only for very large

heating times. In this limit, qualitative agreement between the predictions

of Eq. (5.17) and the experimental measurements is found; the regression rate

increases as both the incident heat flux, Q, and the surface temperature, T

g

,

increase. Likewise, the prediction of Eq. (5.16), that the void fraction,

( 1— 4>) , increases with the temperature of the thermoplastic, coincides with

one's intuitive expectation; the higher the temperature of the medium, the

larger the number of bubbles per unit volume that may nucleate.

-34-
NOMENCLATURE

pre-exponential factor

specific heat

nucleation frequency factor, see Eq. (3.3)

diameter, D = 2R „ 2

-EA8 T

s DamkOhler number, KJ = (a /r )(A/p )e

s ®8

EOtviJs number, E„ = p gD /a o H o o o force; number of liquid molecules per unit volume

gravitational constant

nondimensional surface tension,^ * <j/a

g enthalpy of polymer melt, h = h^^ + Cp(T-TQ

) = E h^Y^

k

enthalpy of component k, h^ = h£®' + cpk(T-TQ

)

vapor-phase enthalpy, hy = h^*^ + CpV (T-TQ

)

k/ cpsT

s nondimensional enthalpy,

overall surface enthalpy

nondimensional value of J [see Section IV. 5]

spatially-dependent component of bubble velocity, J = l/(2y)

Boltzmann constant, k = 1.38054 erg/°K

spatially-dependent component of bubble growth rate,

K = (6/Tr)a(p/pv ) 2 (c

p

AT/,4 ) 2

nondimensional vaporization enthalpy, l = c£./cps

T

s vaporization enthalpy

characteristic conduction length, L * x/pc^r

mass flux

4 3 Morton number, M = gy /pa

r-djl dT

dT J dx

number distribution function

-35-
M.

N

P

P

e

Pr

q

Q

r

A,

R

<£

R

Re

t

T

AT

u

'U

U

k

v

V

wkw

x

nondimensional number distribution function, M. = n/nQ

00 bubble number density, N = / ndR

0

dynamic pressure

nondimensional temperature gradient, p = d0/d£

2 nondimensional pressure,

ff = P/p

g r

Prandtl number, Pr = v/a

heat flux, q = - A + p S \\Y

k

k

incident heat flux

regression rate

nondimensional bubble radius, /l = R/R

q

bubble radius

ideal gas constant

bubble radial growth rate

Reynolds number. Re = 2Rv/v

time

temperature

superheat, AT = T -

velocity of polymer melt

nondimensional melt velocity, 'll = u/r

diffusion velocity of species k in melt

bubble translational ve

4 bubble volume, V = y

ttR

rate of creation of component k in melt

4 2 work of nucleus formation, W = y

iroR = GR

spatial coordinate

mass-fraction of species k in melt

modified spatial coordinate, see Eq. (3.5)

locity, v =1^ , see Eq. (2.3)

-36-
a

8\

r

r

i

6

A

n

0

0

K

X

A

U

V

K

P

p

f

a

TXX

<J> Yv

X

0)

n

k

k

thermal diffusivity, a * X/pc^

nondlmensional activation energy, 6 * E/(J£T

g nondimensional specific heat of species k in melt, = C pk^ C pks

drag-force term, defined by Eq. (4.25)

nondimensional parameters defined by Eqs. (4. 14)-(4. 18) , i = 1,...5

nondimensional overall heat-release factor, see Eq. (5.14)

mass-sink term, defined by Eq. (4.24)

nondimensional melt viscosity, q = y/y

g nondimensional melt temperature, 9 = T/T

g stretched value of 0, 0 = 6 (1 - 0)

nondimensional value of K, k = K/K

s thermal conductivity

Jakob number, A = (p/pv

) (c

p

AT/^jf ), see Eq. (2.6)

melt viscosity

melt kinematic viscosity

nondimensional spatial coordinate, £ = x/L_

m+1 nondimensional value of H , it = II /n R

m m moo 00 m*"*

1 moment of bubble distribution function, n = f nR

m

dR m Jo density (mass per unit volume of material)

fluid density (mass per unit volume of space )

surface tension or surface free energy

shear stress, = - j y , see Eq. (4.8)

density ratio or liquid volume fraction, = P^/p g

» see Eq

vapor-phase density ratio, <f> v

= pv

/pvg

nondimensional thermal conductivity, y = X/X

s nondimensional rate of creation of species k in melt, -

(4.4)

w L

k s

p

s r mass-flux fraction of species k into bubbles

-37 -
Subscripts

b

B

fS

I

o

s

T

v

boiling or saturation condition

bubbles

gas phase

initial bubble growth stage

reference condition

surface

thermal bubble growth stage

vapor

ambient condition

Superscripts

* monomer cluster

-38-
REFERENCES

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(1971).

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-40-
Incident

heat

flux,

h

II H

41

Figure

1.

The

model

configuration.
I

Figure 2. The nucleation and growth of bubbles in the polymer melt. (a) Small

clusters of monomer are formed. (b) The monomer clusters may

nucleate small bubbles, which may grow or collapse. (c) The growing

bubbles translate in the direction of the local temperature gradient;

they are at all times surrounded by a layer of monomer. Bubble

growth takes place through vaporization at the interface.

42
43

Figure

3.

Energy

balance

at

the

surface.
N BS-114A (REV. 2-8C

U.S. DEPT. OF COMM. i. PUBLIC . ,'ION OR 2. Performing Organ. Report No.

1 3. Publ cation Date BIBLIOGRAPHIC DATA

SHEET (See in struction s)

RFPORT NO. NBSIR 85-3130 May 1985

4. TITLE AND SUBTITLE

A Model Describing the Steady-State Pyrolysis of Bubble-Forming Polymers in

Response to an Incident Heat Flux

5. AUTHOR(S)

Indrek S. Wichman

6. PERFORMING ORGANIZATION (If joint or other than NBS, see in struction s) 7. Contract/Grant No. national bureau of standards

DEPARTMENT OF COMMERCE 8. Type of Report & Period Covered

acttSHH«4XH3tt{;xxExx202k4c Gaithersburg, MD 20899

9. SPONSORING ORGANIZATION NAME AND COMPLETE ADDRESS (Street. City. State. ZIP)

National Bureau of Standards

Department of Commerce

Washington, D.C. 20234

10. SUPPLEMENTARY NOTES

Document describes a computer program; SF-185, FIPS Software Summary, is attached.

11. ABSTRACT (A 200-word or less factual summary of most significant information. If document includes a significant

bibliography or literature survey, mention it here)

A theoretical model is developed to describe the in-depth effect of bubbles

on the steady-state transport of volatile gases (monomer) from the surface of a polymer subjected to an incident heat flux. In this model the effect of the

bubbles on the surrounding (liquid) polymer is felt through the bubble number

distribution function, n, which appears in the equations for conservation of

mass, momentum, species and energy in the melt. The equation for the evolution

of n includes the effects of bubble growth, convection and nucleation; its

derivation requires preliminary study of the growth and motion of an individual

bubble in a liquid with a temperature gradient. With these equations, formulas

for the mass flux of volatiles from the polymer surface and the bubble void

fraction are developed, for the special case of constant polymer mass fraction.

12. KEY WORDS (Six to twelve entries; alphabetical order; capitalize only proper names; and separate key words by semicolon s) Bubbles: convection; Heat Flux; Mathematical Models; Nucleation; Polymers;

Steady State

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