a model for intumescent paints

Inl. J Engn8 Sci. Vol. 24, No. 3, pp. 263-276, 1986 Printed in Great Britain.

0020-7225/86 53.00 + .OO Q 1986 Pergamon Press Ltd.

A MODEL FOR INTUMESCENT PAINTS

JOHN BUCKMASTER Department of Aeronautical and Astronautical Engineering, University of Illinois,

Urbana, IL 61801, U.S.A.

and

CHARLES ANDERSON and ARJE NACHMANt Southwest Research Institute, San Antonio, TX 78284, U.S.A.

Abstract-A model is constructed for intumescent paints in which it is assumed that the transition to the swollen state occurs at a very thin zone or front. Across this front there is a discontinuity in the density, the velocity, and the temperature gradient. The temperature is continuous and fixed at the front at a value that is a property of the paint. All of the mass loss through outgassing occurs at the front, and this gives rise to a large increase in the volume of the paint particles as they are traversed by the front. On each side of the front, the physical processes consist exclusively of heat conduction. In the context of this model, a finite coating subject to a heat flux contains two moving boundaries: the front, and the outer surface of the coating at which the flux is applied. Through an elementary mathematical transformation, this problem is reduced to one in which the outer surface is fixed and only the inner front moves. This simplified problem is of Stefan type, easily amenable to numerical analysis. The time-dependent substrate temperature, when calculated in this way, tends to level off at a constant value until the front has completely traversed the coating; then it sharply increases again. This is a characteristic of much of the available experimental data and, combined with visual evidence, lends credence to our model.

1. INTRODUCTION

INTUMESCENT paints are coatings that provide temporary protection against heat for any object to which they are applied. They are used by the U.S. Navy to protect artillery shells and other explosive devices from shipboard fires, and they have been used to protect structures from rocket exhausts. They do this by swelling on exposure to heat to form an insulating layer.

In order for a coating to exhibit tumescence when heated, it must possess two basic properties: It must give off gas, and this gas must be trapped in the form of bubbles, a possibility if the outgassing material is viscoelastic. The conversion of the solid coat to an outgassing viscoelastic substance must occur at quite modest temperatures (a few hundred degrees), so that protection is initiated before excessive heat has passed to the substrate (the shell surface, for example). A typical coating might consist of 23% (by weight) of polysulfide, 3% DMP-30, 22% Epon 828, and 52% Borax.

It might be of interest to note that the physical mechanisms indicated here are responsible for the porous nature of rhyolitic and basaltic magmas, except that outgassing (of water vapor) is triggered by the pressure drop that occurs as the molten magma rises to the earth’s surface [l].

There are two important questions that have to be considered in the development of an intumescent material: What properties should it have to optimize its performance? How can those properties be achieved? The second is the concern of the chemist, who must use empiricism, experience, and intuition in order to answer it. The first falls into the realm of mechanics, and although it also has been answered chiefly by empiricism, there is the promise that mathematical modeling can provide useful insights into the process.

Analyses of this type are limited to two: work done by Clarke et al. at Aerotherm [2], and work done by Anderson and Wauters [3] at Southwest Research Institute. Both of these studies formulate equations for mass and energy conservation, account for the tumescence by means of a simple model that relates its extent to the mass loss, and construct numerical solutions.

t Present address: Hampton University, Hampton, VA 23668, U.S.A.

263 ES 24:3-A

264

1.0

0.8

0.8

I

3

0.4

0.2

0

J. BUCKMASTER er ul.

Temperature, “C

Fig. 1. Mass-temperature history for polysulfide.

It is the manner in which the tumescence is modeled that distinguishes this paper from earlier work. If the components of an intumescent paint are heated in an oven in a carefully controlled fashion, they typically exhibit significant mass loss over a narrow temperature range; Fig. 1 shows the mass-temperature history of a polysulfide sample. The temperature at which this loss occurs differs for different components, so that the paint itself exhibits mass loss over a wide temperature range (Fig. 2). This feature is a part of the existing models; and since they relate mass loss to tumescence, the latter occurs over a wide temperature range.

To the contrary, examination of coatings that have been cut through? after partial tumescence suggests that, in practice, tumescence occurs over a region that is thin in comparison to the coating thickness, corresponding to a small temperature range. There is no contradiction between this observation and the results for mass loss in ovens. As we have observed, mass loss by itself is not sufficient for tumescence; the material must be in the right state to trap this gas. The implication is that a proper viscoelastic state and outgassing are only simultaneously achieved over a narrow temperature range. It is the recognition of this possibility that distinguishes this study from earlier ones. Indeed, we assume that the temperature range is so small that tumescence is confined to a front of negligible thickness whose temperature is a prescribed property of the material. This front travels through the coating from the free surface to the substrate as heat is supplied, leaving behind swollen material.

This simple model has two advantages. Not only is it consistent with the experimental observation, but, in addition, it eliminates the influence of modeling assumptions about the least-understood aspect of the whole problem-namely, the tumescent process itself. The mass and volume changes that occur across the front must be specified, but these are fundamental parameters that can be determined from experiment.

Time variations of substrate temperature have been determined experimentally for a large number of coatingst In Section 6 we make qualitative comparisons between these results and the predictions of the model; these provide good evidence that the frontal model is realistic.

t We have done this for coatings of initial thickness 1 cm or so. t These were prepared at Southwest Research Institute, and the tests were performed by personnel at the

Naval Air Development Center, Warminster, PA, U.S.A.

Model for intumescent paints

I 1 I 1 I I I I I I

I

265

1.00

Boo

.600

NM

.200

Temperature, “C

Fig. 2. Mass-temperature history for a sample of intumescent paint.

2. THE MATHEMATICAL MODEL

We consider a one-dimensional configuration, as shown in Fig. 3. In due course, all source terms, whether of mass, volume or heat, will be taken to be delta-functions, consistent with the frontal model, but our initial discussion will not make this restriction. Unlike the Lagrangian formulations of the earlier analyses, we shall adopt an Eulerian description.

Outgassing causes mass loss at a rate g, so that the equation for mass conservation has the form

(2.1)

where p is the density and u the velocity. Consider an isolated mass of virgin material, and consider what happens as it is

heated. The increase in temperature turns the paint into a viscoelastic fluid that gives off gas, most of which escapes. The generation of this gas creates voids that become frozen

x=0 X

u=o

Virgin Material

= h(t)

u = u(t)

Swollen Paint

Energy

Flux

Backing Plate

Front Outer Surface

Fig. 3. Configuration.

266 J. BUCKMASTER et al.

into the material as it hardens as more of the gas is liberated. Thus mass is lost, but, at the same time, the volume increases. The earlier analyses [2, 31 adopted a functional relationship between these changes of the form

AVIV = f(Amlm;), (2.2)

where Am, AI’ are the mass and volume changes, and mi and Vi are the initial mass and volume. It is a virtue of the frontal model that such poorly understood relationships are not required.

In order to construct a field equation that describes these changes, we use the concept of Lagrangian mass, m. Consider the mass of a unit volume of virgin material-it has a volume pi_ We now follow all of the particles in this volume, deleting those that turn into gas. The total mass is m and it decreases from the initial value pi due to outgassing; its volume is V. Since, according to (2.1) g is the rate of mass loss per unit volume, it follows that

(2.3)

The mass and volume are related through the formula

m = pV. (2.4)

The only significant energy is thermal in nature, so that the energy equation has the form

where we suppose that C, is a constant throughout the paint. Q is a measure of the energy lost due to outgassing; this lost energy is proportional to i. In view of the continuity equation (2. l), the energy equation may be written as

If energy is lost simply because of the removal of mass from the system, and enthalpy changes intrinsic to the change in phase are negligible, then C’,T - Q is zero, and the source term in (2.6) vanishes.

The field equations are completed by the specification of g. The quantity g/p is the rate of mass loss per unit mass and, plausibly, is a function of T and m - q, where mf is the final Lagrangian mass, a specified quantity. A possible choice is

.ij = pDe-‘IT(m - mf)“, a! 2 0, (2.7)

but for the frontal model no explicit choice is necessary. To complete the mathematical description, appropriate initial and boundary conditions

must be imposed. We shall restrict our discussion to the case where T is initially constant everywhere in the coating:

t = 0: T= To, (2.8)

a constant flux is applied at the outer boundary (x = L):

x = L(t): k(dT/dx) = E, (2.9)

Model for intumescent paints 26-l

and the paint is applied to a thin, highly conducting plate of density ps, specific heat C,, and thickness d,, so that

x = 0: k g = p&ds $ . (2.10)

These conditions approximate those of experiments whose data we wish to compare with the theoretical results.

3. THE DELTA-FUNCTION MODEL

The equations described in Section 2 are unsatisfactory in several respects-certainly the need to adopt formulas of the form (2.2) and (2.7) is unfortunate. We shall bypass these flaws by adding the additional assumption, one for which there is experimental evidence, that tumescence only occurs in a very thin zone. From this point of view there are two types of mass loss: that which leads to tumescence, and that which does not. The former is associated with a g that is nonzero only at one temperature (that of the zone or front) and is large at that temperature, so that its spatial integral is significant. This is the only type of g that we shall consider-the mass loss that occurs elsewhere (recall Fig. 2) is less important, only generating changes in thermal capacity for the most part, and is neglected.

With this assumption a coating subject to a constant heat flux is divided into two regions separated by the front. Between the substrate and the front there is stationary material with uniform density. The temperature in this region is everywhere less than that at the front. Between the front and the free surface the material also has a uniform density, smaller than the initial value, as a result of processing by the front; this density does not change with time. The temperature in this region is higher than the front temperature, and the velocity is spatially uniform but nonzero and varies with time. The configuration is shown in Fig. 3.

The only nontrivial equation on each side of the front is therefore the homogeneous energy equation

(3.1)

where appropriate uniform values for p, u and k must be assigned for the two regions. Connection or jump conditions across the front are deduced by an analysis of the front structure.

The location of the front is defined by

x = h(t), (3.2)

and to examine its structure we introduce the new variable [ by means of the formula

x = h + @; (3.3)

d is a small parameter that characterizes the thickness of the front. It is eventually set equal to zero.

The governing equations of Section 2 are now rewritten in terms of the independent variables t and t, and, at the same time, expansions of the form

T = T, + 8~ + o(S2), (m, P, 4 - h P, u) + o(@. (3.4)

are adopted. T* is a constant, since there is ,no significant change in temperature as the front is traversed, and m, p, u now stand for the leading terms in expansions in 6. The


continuity equation (2.1), eqn (2.3), and the energy equation (2.6) can now be written. to leading order, as

& (pu - p/z) = -s.$

Comp~ng eqns (3.5) and (3.6), we have

(3.5)

(3.6)

(3.7)

(3.8)

which may be integrated to yield

m = C(t)(pu - pk). (3.9)

The solution within the front must match with the outer solutions on each side, so that eqn (3.9) must be consistent with the conditions

.$-+--GO: m - ml, P - Pi, 24 - 0,

~--+-l-CO: m ---) mf, P 4 Pf? u - q(t); (3.10)

here the subscript i refers to the initial virgin state; the subscript frefers to the final state after tumescence has occurred. In this way we deduce the result

(3.11)

which relates the instantaneous speed of the coating behind the front to the instantaneous speed of the front itself.

A similar treatment for the energy equation (3.7) leads to the condition

h(Q - C,T, )[m], (3.12)

which, together with,

WI = 0, T= T,, (3.13)

completes the specification of the connection conditions. Here the square brackets denote evaluation on the processed side of the front minus evaluation on the virgin side.

4. NONDIMENSIONAL FORMULATION AND REDUCTION

TO A STEFAN PROBLEM

The problem of solving the energy equation (3.1) on each side of the front, together with the connection conditions (3.1 I)-(3.13) and conditions (2.8~(2. IO), is a generalized Stefan problem-generalized in the sense that there are two boundaries whose locations must be determined as part of the solution: namely, the front and the outer surface. In this section we shall formulate an equivalent nondimensional problem and then, by means of an elementary transformation, reduce it to one with but a single unknown surface.

Model for intumescent paints 269

We shall consider the response of the paint to a constant heat flux. Initially the coating has a uniform temperature, less than T *, and thickness d. The heat flux raises the temperature, and after a time ti the surface temperature reaches the value T*, signaling the onset of tumescence. We shall call this initial phase, occupying the interval (0, tl ), the preheat phase.

Continued application of the heat flux now causes the front to move into the interior of the paint. This continues until the front reaches the substrate at a time t2. We shall call the interval (t,, t2) the tumescent phase. At the end of it the paint has thickness D.

The final, or post-tumescent phase, is similar to the preheat phase in the sense that it is characterized simply by an increase in temperature. The coating is, of course, thicker, and its physical properties are different.

Consider the tumescent phase. During this period the outer front moves a distance D - d at a speed z+(t), and the front moves a distance d with speed h. Thus,

t2

D-d= s ‘f2

uf dt, d= - J

i? dt. II 11

It follows from eqn (3.11) that

D pimr -=- d Pfmi ’

so that (3.11) may be written as

uf= -[(D/d) - l]h.

The location of the outer (free) surface is then

x = L(t) = D - h(t)[(D/d) - 11.

Since mi = pi and mf = pfD/d, the jump condition (3.12) becomes

[ Tl k$ =h(Q-CpT*)(pfz-pi).

(4.1)

(4.2)

(4.3)

(4.4)

(4.5)

In order to nondimensionalize the equations, we shall use d for the characteristic length, p&d2/kf for the characteristic time. The nondimensional temperature is (T - To)/(T* - To), and the characteristic energy flux used to nondimensionalize E is ki(T* - To)/d. At the same time it is convenient to define by the formulas

Pik. a=- Pfki ’

fi=k’ kf’

,=E Pf ’

6=

D CT=-, Q - CJ,

d P = a(a - Y)

C,(T, - To) .

With T, x, t, h, L and E now standing for nondimensional be written in the form

O<x<h: aT d2T

adt=s’

nondimensional parameters

h<x<L: $? 1)6g=g.

(4.6)

variables, the problem may

(4.7)

(4.8)

During the preheat phase, h = L = 1; during the post-tumescent phase, h = 0, L = 6. Boundary conditions at the front are


T= t, Bi,,,-p~lh_o=~~. (4.9)

At the surface of the substrate to which the paint is applied, condition (2.10) becomes

(4.10)

The first of these is correct until the post-tumescent phase, whereupon the second is appropriate (the conductivity changes).

At the outer surface, (2.9) becomes

x = L: dT

-E or z- (4.11)

the first of these is correct during the preheat phase, the second at later times (again, due to the conductivity change).

The initial condition becomes

T=O at t = 0. (4.12)

Consider now the new variable defined by the formulas

s=x for OG.X=G/Z, s=X+b-lh@) for h<x<L. (4.13) u cr

This fixes the outer surface at s = 1, and eqns (4.7), (4.8) become

O<S<h: dT 8T

a-g=-@ h<s<t: ,dT a2T

u y-as’3

with front conditions

7th t) = 1, ~1 _ dT - CT@--- I a.9 s=.+O

= ph. s-h+0

The boundary conditions at s = 0 are

dT = 6 CT dT dT

a~ at (OGt<t2), -g=u/36-$ (t > f2L

andats= 1,

dT E -= as

(Oct-ct,), $=cq3E 0 z=- tl)-

(4.14)

(4.15)

(4.16)

(4.17)

Equations (4.12)-(4.16) define a free-bounda~ problem of Stefan type that can be solved numerically. It differs from the common type of Stefan problem only in the fact that the front motion is irreversible, but this plays no role for the specific initial-value problem treated here. We shall describe some solutions in Section 6.

5. ANALYSIS

5. I. The preheat phase: ~~u~in~ of the virgin rn~eriu~ During this phase, before tumescence has occurred, it is necessary to solve (4.i4a)

subject to the boundary conditions (4.16a), (4.17a) and the initial condition (4.12). The solution obtained, using a Laplace transform in t, is


y-=x- 27ri srQd s

( 1 /P) co& (\J@P)s) + [ Wkcup)l sinh (~P)s)

v( cup) sinh ~(I(LY~) + Sp cash f(orp) . (5.1)

This is not a useful representation for finite time, but it does provide a description for large times, specifically

Et T--

1 Ecu

a+6+2a+6 --~~+-&~-E~(ai++~;~+o(l).

cy (5.2)

For typical parameter choices this late time solution is valid before tumescence starts. A description for finite time is best obtained by numerical means. The procedure

adopted was the method of lines described by Meyer [4], since it is by far the best technique for dealing with the tumescent phase. To this end, implicit differencing in time leads to the equation

dzT, dx*

-;T.= -;T,,, (5.3)

where T, is the temperature evaluated at time t,. The problem is now imbedded in a one-parameter family satisfying eqn (5.3) together with boundary conditions

T,(O) = B, T’,(O) = (WAt)[B - Tn-I(O (5.4)

the parameter is B. For one choice of B, initially unknown, this defines the solution to our problem. In view of the linearity of the field equation, we may write

T:, = PT,, + Q, (5.5)

where the functions P, Q are independent of B. Substituting into (5.3) and using the boundary conditions (5.4) then leads to the initial value problem

P’+p*-;=o, Q’ + PQ = - ; T,-,,

P(O) = :t 9 Q(o)=_&.$, (5.6)

which is easily integrated by using the trapezoidal rule. At the outer surface the temperature is

T,(l) = (E - MlNIP(lh (5.7)

and this provides the initial condition for the integration of (5.5) back towards the origin. This completes the determination of T,, so that time can now be advanced and the procedure repeated.

5.2. The tumescent phase: propagation of the front

During this phase it is necessary to solve eqns (4.14)-(4.17). The treatment of the virgin material proceeds as in Section 5.1, up to and including the calculation of P and Q. The solution of (4.14b) between the front and the free surface proceeds in a similar fashion, so that writing

T:, = RT,, -I- S (5.8)


leads to the initial value problem

RfR2-$0, s’+RS=-;T,_,,

R(1) = 0, S(1) = a/3E, (5.9)

These equations are also integrated by using the trapezoidal rule. With the functions P, Q, R and S known, the position of the front can be determined.

The energy condition (4.15b) implies that, at the front,

R(h,) + S(h,) - aDIP + Q&)1 - WAWn - hn-11 = 0: (5.10)

it is a simple matter to determine where this function vanishes. Once the front location is calculated, eqns (5.5) and (5.8) are integrated away from

the front, using the condition there that T = 1, and in this way T, is determined everywhere.

5.3. The post-tumescent phase This is described by eqn (4.14b) with boundary conditions (4.16b), (4.17b). We solve

for R and S as in Section 5.2 and then integrate eqn (5.8) subject to the initial condition

T,(O) = WV@t/&O + Tn- I (0)

1 - R(O)(At/@) * (5.11)

The late time solution is

PE T-pb++

&3E

2(@ + a) s2 +

UP%E - s + constant + o( 1). ps + u

(5.12)

The rate of increase is bigger than during the preheat phase, since the coating has lost mass and so has a diminished thermal capacity. This draws attention to an important point about intumescent paints: They work in the context discussed here not because one ends up with an insulating layer- if the energy flux is fixed, that is irrelevant-but because of the unsteady processes associated with the formation of that layer. The outward flux of material associated with the thickening tends to hold the heat in the outer portion of the layer, delaying its passage to the substrate. In addition, energy is absorbed at the front.

6. NUMERICAL RESULTS AND COMPARISON WITH EXPERIMENT

An important point to make in our discussion is that the physical evidence for a frontal model is only suggestive, not convincing. In constructing numerical solutions then, we seek qualitative features associated with the frontal model whose duplication or absence in experimental data will confirm or refute the model; detailed quantitative comparisons are not our goal at this stage of the modeling process, and accurate reliable experimental data is not available. We adopt parameter values consistent with the experimental data of reference [3]; dimensional values are shown in Table 1. The choice of T, can only be justified after solutions have been constructed, as we shall see.

Table I.

Ps Pi c~s C, d, d D ki kr TO Ta E

7.8 1.49

gm/cm)

0.1 0.4

cal/gm-“C

.08 .I

cm

.45 5.5 x 1o-3 2 x lo-’ 20 150 2.68

Cal/cm-s-“C “C cal/cmz-s


Two items are missing from Table 1. One is the heat loss parameter Q whose value determines CL [eqn (4.6)]. We shall construct solutions for various values of I.C. The second missing parameter is p,-, the final density of the coating. The final mass of the physical sample was four-tenths of the initial mass, so that, in view of the 4.5-fold increase in thickness, this corresponds to a final density equal to one-eleventh of the initial value. Most of this mass loss occurs in the temperature range 275-325°C well behind the front, which has a temperature (T* ) of 150°C. Since the mathematical model only accounts for mass loss at the front, it might be more appropriate to consider only the mass loss up to the temperature 150°C. Only one-tenth of the mass is lost at that temperature that, with the same volume expansion of 4.5, corresponds to a density one-fifth of the initial value. Because of this, we shall consider values of y in the range

y - 5-11.

The other nondimensional parameters inferred from Table 1 are

E .4, 4.5, /3 2.75, 6 Y = (r = = = CY, a=- - P

1.8-4,

the uncertainties in (Y arising from those in y. It is noteworthy that

kilkf= p N (D/d)2’3.

This is exactly what is predicted by dimensional reasoning if the decrease in conductivity of the swollen paint is attributed simply to a decrease in effective conducting area due to the voids created by the liberated gas.

Figure 4 shows the variations in substrate temperature with time for three different values of (Y, with P = 0. Corresponding variations of h are shown in Fig. 5. During the tumescent phase (1 > h > 0), the temperature rises but tends to level off at the value 1. It cannot increase beyond this value until the front has reached the substrate and tumescence is finished with. This leveling-off occurs even though there is no energy loss at the front; it is associated with the temporary retention of energy in the increasingly thick region between the front and the free surface. Energy is trapped there, its passage

Time

Fig. 4. Wall temperature vs time for p = 0, a = 1.8, 3, 4.

274 J. BUCKMASTER et a/

0.6

Time

Fig. 5. Front location vs time for p = 0, o( = 1.8, 3, 4.

to the substrate delayed, because the convective flux associated with the expansion opposes the conductive flux. Experimental substrate temperatures reported in (31 display the same characteristic (Fig. 6) and it is this that justifies the value chosen for T*. In this way the front is identified, for the paint at hand, with the region in which bound water is driven off from the constituent borax.

Heat loss at the front enhances the size of the step in the temperature curve. This is apparent from Fig. 7, which shows the response for different values of P when a! = 4. Also shown is the response in the absence of tumescence, when the preheat phase is extended indefinitely. It is clear that for a fixed-energy-flux problem there is no advantage to tumescence if the step is not significant.

A wide variety of intumescent paints has been prepared at Southwest Research Institute and tested at NADC, and all of these display steps in their responses (Fig. 8).

““I

00 0.6 1.0 1.6 2.0 2.5 3.0

Time Mwtesl

Fig. 6. Experimental variations of wall temperature with time from reference [3].


0.8

Time

Fig. 7. Wall temperature vs time for a = 4, p = - I, -5, -20, -50 (also shown is the response in the absence of tumescence).

400-

Y

f

[” c”

mo-

*

I , I , I , I / I

1 , I !

B-B_ c

. . . . . . . . . (-J

-w- E

0 1 .o 2.0 3.0 4.0 5.0 6.0

Time IMinutef.1

Fig. 8. Experimental variations of wall temperature with time for a number of different paints. E is the response of a nontumescent material; B shows evidence of two fronts.


(The curve labeled B has a double step, corresponding to two fronts). We regard this as strong evidence that a frontal model contains the key ingredient necessary to understand the behavior of these paints. One difference between the experimental and theoretical results is in the curvature following the step-the theoretical temperature sharply increases again once the front has reached the substrate, in contrast with the more gradual increase displayed in the experiments. This could be due to the finite thickness of the real front or the mass loss that continues to occur behind the front (this loss can be endothermic).

7. CONCLUDING REMARKS

There is strong evidence that the tumescent process in intumescent paints occurs in a narrow region or front. In the case of thick coatings this evidence is partly visual-a cut coating shows what appears to be a sharp transition. This is only really clear when seen in color, so that we are unable to display the evidence in these pages. The rest of the evidence is in the numerical solutions constructed here, which display a characteristic step in the wall-temperature history. This history agrees much better with experiment than the solutions of [3], obtained under the assumption that the tumescence is distributed.

The tumescent process is endothermic, and this can add significantly to the protection provided by the coating, but it is not an essential ingredient for the step; all of the solutions shown in Fig. 4 are for adiabatic tumescence.

A more realistic model would account for the mass loss (without additional tumescence) that occurs between the front and the free (outer) surface, a necessary addition if quantitatively accurate predictions are hoped for.

Acknowledgments-This work was performed under U.S. Navy Contract N62269-81-C-0246 for the Naval Air Development Center (NAM)), Warminster, PA. In addition, the first author was partly supported by the National Science Foundation and the U.S. Army Research Office. We wish to thank Mr. Dave Pulley at NADC and Mr. Paul McQuaide at the Pacific Missile Test Center, Point Mugu, CA for their active interest and support. Also, we wish to acknowledge Mr. William Mallow and Mr. Jerry Dziuk at Southwest Research Institute for their insights into the chemistry of intumescence and their innovative formulations developed for this program.

REFERENCES [ 1] R. S. J. SPARKS, J. Volcanology Geothermal Rex 3, 1 (1978). [2] K. J. CLARKE, A. B. SHIMIZU, K. E. SUCHSLAND and C. B. MOYER, Aerotherm Final Report 74-101

prepared for NASA, June ( 1974). [3] C. E. ANDERSON, JR. and D. K. WAUTERS, In?. J. Engng Sci. (to be published). [4] G. MEYER, Siam Rev. 19, 17 (1977).

(Received 2 1 Seplember 1984)

a model for intumescent paints

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