A Finite Element Analysis on the Modeling of Heat Release Rate, as Assessed by a Cone Calorimeter, of Char Forming Polycarbonate

David L. Statler Jr. & Rakesh K. Gupta

Polymer Research CenterDepartment of Chemical Engineering

October 10, 2008

COMSOL Multiphysics Conference, Boston, 2008

Presented at the COMSOL Conference 2008 Boston

Outline• Introduction

– Cone Calorimeter– Heat Release Rate Data

• Modeling of Heat Release Rate– Defining the Problem– Solving the Problem– Results

• Conclusions • Acknowledgment

Cone Calorimeter

Adapted from http://www.doctorfire.com/cone.html

Performed at Marquette University

Measures– Time to Ignition– Heat Release

Rate (HRR)– Peak HRR– Mass Loss Rate

Relates HRR to O2 consumed

HRR vs. Time

-100

0

100

200

300

400

500

600

0 100 200 300 400 500

Time (sec)

HR

R (

kW

/m2 )

a

PC no FRPC Br FRPC KSS FR

HR

R (k

W/m

2 )Cone Calorimeter Data

Radiant Heat Flux: 50 kw/m2

Lower Time to IgnitionLower PHHR

±10% Error

Cone Calorimeter

Adapted from http://www.doctorfire.com/cone.html

0.E+00

1.E+05

2.E+05

3.E+05

4.E+05

5.E+05

6.E+05

0 50 100 150 200 250 300 350 400Time (sec)

HR

R (

W/m

2 )

Time to Ignition

PHRR

Plateau caused by char growth

PC No FR50 kW/m2 Heat Flux

Polymer Plaque

Remaining Char

Cone Calorimeter

Cone Heater

Polymer Sample

Cone Calorimeter

Cone Calorimeter & Polymer Combustion

Polymer -Long Chain Molecules

Heat

PyrolysisPolymer Breakdown

To Short Chain & Small MoleculesPolymer -

Long Chain Molecules

FR in Condensed PhaseProtective Char Formation:

Barrier to Heat and Mass Transfer

Polycarbonate is a Natural Char Former

O C OO

CCH3

CH3

n

O C O

O

CCH3

CH3

n

Cone Heater

Mathematical Modeling of Heat & Mass Transfer, along with Char Formation, to predict Heat Release Rate from Cone

Calorimetry Data

Cone Heater

Polymer Plate (3mm thick)

z

xy

Combustion (% Heat of Rxn fed back to Boundary)

Char Formation Zone (very thin, then grows)

3 Dimensional Transient

1 Dimensional Transient

Radiant Heat

Convective Heat Lostx0

Insu

latio

n Polymer Zone

xx1, x2

Polymer Zone: Mass Transfer

• Polymer Pyrolysis

• 1st Order Rate Expression

• Assume Constant Volume

CGPor

CharGasPolymer

k

k

⋅−+⋅→

+→

)1(0

0

αα

pp mk

tm

⋅−=∂

∂0

⋅−

⋅=TR

EAk A0

00 exp

pp

P ckt

cr ⋅−=

∂

∂= 0

Polymer Zone: Mass Transfer

• Gas Generation Rate Expression

• Gas Expression

pG

polymerG

PG ckxc

Dt

cr ⋅⋅=

∂∂

⋅−∂

∂= 02

2

α

Char Zone: Mass Transfer

02

2

=∂

∂⋅−

∂∂

xc

Dt

c Gchar

G

Polymer Zone: Heat Transfer

Char Zone: Heat Transfer

Ppolymerpolymerpolymer ckHxTk

tTCp ⋅⋅∆−=

∂∂

⋅−∂∂

⋅⋅ 002

2

ρ

02

2

=∂∂

⋅−∂∂

⋅⋅xTk

tTCp charcharcharρ

Char Growth

• Rate of Char Production

• Rearrangements

• Defining Mesh Velocity (smoothed step change)

PPC mk

tm

tm

0)1()1( ⋅−=∂

∂⋅−−=

∂∂ αα

txA

tAx

tV

t

m

tmLHS Char

C

C

CharChar ∂∂

=∂

∂=

∂∂

=∂

∂

=∂

∂=

ρρρ

1

PChar

P

Char

mkt

mtxA

ρα

ρα 0)1()1( ⋅−

=∂

∂−−=

∂∂

χρχα

χρχα

⋅⋅⋅−

=∂

∂⋅⋅

⋅−−=

∂∂

Amk

tm

Atx P

Char

P

Char

0)1()1(

PChar

P

Char

ckt

ctxmvel

ρχα

ρχα ⋅⋅−

=∂

∂⋅−−=

∂∂

= 0)1()1(

0

)()1(

x

P

Char tcmvel∂

∂−⋅

⋅−=

ρχα

00

)()1()min,(1x

P

CharxPpolymer t

cscalechsflcmvel∂

∂−⋅

⋅−⋅−−=

ρχαρ

Variables Solved For:

• T(x,t)• cP(x,t)• cG(x,t)

• CharSurface

G Dx

cHHRR ⋅∂

∂−⋅∆= )(1

Temperature Profiles

Concentration Profiles

Heat Release Rate

Initial & Boundary Conditions

Initial Condition (x,0)

ZonePolymer [x0-x1] Char Formation [x1-x2]

cP cP = ρpolymer cP = 0

cG cG = 0 cG = 0

T T = Tinitial T = Tinitial

Boundary Condition (x0;x2,t)

Boundary

x0 x2

cP Flux=0 Flux=0

cG Flux=0 cG=0

T Flux=0 ( )441 )( TTD

xc

HFlux coneCharG −⋅⋅+⋅

∂∂

−⋅∆⋅= σεφ

Equations are Coupled

• No Analytical Solution Exists• Numerical Solution Sought

– Finite Element Method– COMSOL Multiphysics®

• Solve Simultaneously System of Partial Differential Equations

• Char Growth Simulated by Moving Boundary

HRR Results; PC No FR

0.E+00

1.E+05

2.E+05

3.E+05

4.E+05

5.E+05

6.E+05

0 50 100 150 200 250 300 350 400 450 500 550 600Time (sec)

HR

R (W

/m2 )

Cone Data

COMSOL Results

HRR Results; PC Br FR

0.E+00

1.E+05

2.E+05

3.E+05

4.E+05

5.E+05

6.E+05

0 50 100 150 200 250 300 350 400 450 500 550 600Time (sec)

HR

R (W

/m2 )

Cone Data

COMSOL Results

HRR Results; PC KSS FR

0.E+00

1.E+05

2.E+05

3.E+05

4.E+05

5.E+05

6.E+05

7.E+05

0 50 100 150 200 250 300 350 400 450 500 550 600Time (sec)

HR

R (W

/m2 )

Cone Data

COMSOL Results

Modeling Conclusions

• Model predicts Heat Release Rate for Char Forming Polycarbonate with and without Flame Retardants

0.E+00

1.E+05

2.E+05

3.E+05

4.E+05

5.E+05

6.E+05

0 50 100 150 200 250 300 350 400 450 500 550 600Time (sec)

HR

R (

W/m

2 )

Cone Data

COMSOL Results• All parameters in model can be found experimentally, such as, kinetics by TGA

Acknowledgments• George and Carolyn Berry (Berry Fellowship)• WV EPSCoR Research Challenge Grant Program (Funding)• Department of Energy (Funding)

• Dr. Pierre Moulinie from Bayer (Makrolon® Polycarbonate)• James Innes from Flame Retardant Associates (Sloss

Industries KSS FR®)• Dr. Sushant Agarwal from WVU (Research Assistant Professor)• Dr. Charles Wilkie from Marquette University (FR Expert)• Dr. Carlos Hilado from Product Safety Corp. (FR Expert)

Thank you!