A Finite Element Analysis on the Modeling of Heat Release Rate, … · A Finite Element Analysis on...
Transcript of A Finite Element Analysis on the Modeling of Heat Release Rate, … · A Finite Element Analysis on...
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!