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NATIONAL BUREAU OF STANDARDS REPORT
964 6
A REVIEW OF IKE STATUS OF
FIRE PREFORMANCE PREDICTIVE METHODS
Stanley P Rodak; Physicist, Heat
U.S. DEPaRTMErr OF COMMERCE
NATIONAL BUREAU OF STANDARDS
-
THE NATIONAL BUREAU OF STANDARDS
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-
NATIONAL BUREAU OF STANDARDS REPORT
NBS PROJECT
421 5122
NBS REPORT
November 21, 1967 9646
A REVIEW OF THE STATUS OF
FIRE PfiEFQRllANe-E PREDICTIVE METHODS
by
Stanley P. Rodak; Physicist, Heat
IMPORTANT NOTICE
NATIONAL BUREAU OF STA^for use within the Government. B<
and review. For this reason, the p
whole or in part, is not authorize
Bureau of Standards, Washington,
the Report has been specifically pr
Approved for public release by the
director of the National Institute of
Standards and Technology (NIST)
on October 9, 2015
accounting documents intend^
bjected to additional evaluation
sting of this Report, either in
)ffice of the Director, National
he Government agency for which
ies for its own use.
U.S. DEPARTMENT OF COMMERCE
NATIONAL BUREAU OF STANDARDS
-
A REVIEW OF THE STATUS OF
FIRE PREFORMANCE PREDICTIVE METHODS
by
Stanley P. Rodak; Physicist, Heat
The problems of fire-resistance have attracted the attention
ofvarious workers. This project is concerned with the theoretical
aspectsof fire resistance. From a knowledge of the thermal
propertities of abuilding material shaped into a particular
geometry, we wish to be ableto predict the thermal fire rating**of
scaled constructions of the samematerial. Ideally, we would like
the calculations techniques to includevariable thermal properties
and endothermic and exothermic processes.
In order to circumvent certain numerical difficulties in using
vari-able thermal properties, it was felt that sufficient
information wouldbe had if the calculations are made in a manner
that will allow the infor-mation to be displayed in the following
fashion. Giedt (1) has suggested
01'J
\U
r
)
Alii
ino
'4- =
LD
u4.’>
» F F \ . r 'if
a mean value for thermal conductivity when it varies with
temperature.
Endothermic and Exothermic processes are to be disregarded
untilthe numerical techniques are sufficiently developed.
The mathematical equations to be used are as follows:
parabolic equation of heat flow* -Q||) T = 0 (1)
boundary condition ,St
^Sn= -hT (2)
(n is the normal surface component)
* Nomenclature is at end of text.** The thermal fire rating will
be taken as the time before the tem-
perature in a region in the specimen under consideration
reachesa value predetermined to be unsafe for fire-resistance
purposes.
-
2
The mathematical model which we will work with is a
homogeneoussolid bounded by two parallel planes. The problem will
be two-dimensional.
Over one surface, the exposed surface, we define a temperature
riseby
T(o,y,t) = f^ (t) (3)
where f^ (t) is the ASTM time-temperature curveo
On the other surface, the unexposed surface, the heat flux is
thatfor a horizontal surface (2,3)
h = h + h (4)n c
h = (.174e/(T -T )) • ((T /lOO)^ - (T /lOO)^) (5)XI so s o
1/3h = .275 T
'
(6)c s
For a high, vertical surface, h c would still be proportional to
Tbut the coefficient changed (3,4). The mechanism of
transpirationcooling was not considered (5).
The numerical technique we choose should be accurate enough to
calcu-late temperatures for our test example (6, pg 126).
TEST CASE:
The region o
-
3
The first studied was the forward-difference explicit algorithm
(8);
pcAx*" (T. .^,3
- .) = kAt (T^ + - 4T^ .)i^j 1-1J 1+1; J i;J-l ^^J+^ (8)
Here, AY .. AX
For a slab NAX thick and MAY tall,
Tj = f (kAc) J = 1, 2, ..., m (9)j
t lxis the temperature on the exposed surface at the k interval
of time*.
The requirement that At /Ax“ be chosen such that
kAt ^ IpcAx^ 4 (10)
in order to have stability, that is, the finite-difference
algorithm isimmune to the accumulation of round-off errors (7),
caused a survey ofother methods (metal rods were to be imbedded in
the test example interior).
The backward difference implicit algorithm (8) is defined by
pcAx^ (T^'^JJ J
.) = kAt (Tk+1
i-1 ,3+ Tk+1
i+1, j+ T
k+1i,j-l i,j+l
k+1- 4t7!) (11)
This method is stable for all values of At/Ax'". Iterative and
matrixreduction techniques are available to solve the resulting
unknown equa-tions (9,10,11,12). For a solid of N interior points
in each direction,
equations result for the two-dimensional problem.
The backv/ard-dif ference implicit algorithm was applied to a
parti-cular problem: finite thick slab exposed on one surface to
temperature
*Ic Ic
To meet the boundary condition (2), we replace (T . ^ . - T, , )
by
Xj -k i
T^ j) by 1/2 in equation 8. The unexposed surface is at i = +
1;
i 1 is the exposed surface, we make a similar replacements in
(11)and (12) to meet boundary conditions of (2).
-
4
unexposed surface has constant "cooling” coefficient ho,
metalrods were placed 1" from exposed face. The equations were
solved foreach successive time increment (At = 1 min.. Ax = Ay =
.2") by GaussSiedel Iteration*. Figs. 1 & 2 are graphs of the
computer solvedproblem
.
Fig, 2 contains two different profiles of the distance from
exposedslab surface versus temperature. One profile is taken along
a line (AA
'
)
normal to the slab surface and passing through one of the steel
rods.The other is along a line (BB*), parallel to AA', but about 3"
from thesteel rod (the steel rods imbedded in the concrete slab
were on 6" centers)Temperatures at 30 min., 50 min., and 210 min.,
intervals along the lineAA ' from 0 to .8" have negative curvature
(13). This is not the correctslope ( 14)
.
The Crank-Nicolson algorithm (15) is reported to be
unconditionallystable (16). It is obviously based on sounder
physical arguments thanthe two previous methods. The algorithm
is
. 2 /„k+lpcAx (T , ,
J
t; ) = (kAt/2)J
(
k+1 k+1 k+1 k+1^'i-1, j
^ \+l, j ^ j-1 ^ j+1^
T^ + T*f . + T^ , T + T^f - 4T^"^^ - 4 t'!^ .)1-1, j 1+1, j 1,
j-1 i,j+l i,j i.j( 12 )
The same test case (metal rods imbedded in a slab as used for
the backwarddifference implicit algorithm was used to test this
algorithm. Graphs ofthe computer solved problem are displayed in
Fig. 3 & 4 . The distance-temperature curves in Fig. 4 have the
proper curvature.
As a precursory check of the Crank-Nicolson algorithm, it was
usedto calculate surface temperatures of the test example
(homogeneous solidbounded by parallel planes). The resulting
surface temperature calcula-tions, as well as those predicted b}’'
theory, are displayed in Fig. 5.The surface temperatures differ by
307o at 100 minutes.
In order to make the Crank-Nicolson difference approximation
moreclosely approximate the differencials, equations were derived
that allowthe Ax spacing to vary in an unequal manner through the
slab; the spacingAy is equal. This would allow a fine grid near the
unexposed surface,where the temperatures vary more slowly with
time. Note 2 discusses howa grid spacing was chosen for .17,
"accuracy."
Another problem was encountered, for the above example used as
acheck in the variable Ax computer program, interior temperatures
werehigher than surface temperatures , k i nr. i i • i o
(T = 100 F, k = I, . .. ; J = I, 2, .
J
* The equation showed diagnol-dominance, hence the choice of
Gauss-Siedel Iteration techniques. See Note 1 for termination
procedures.
-
5
1 2The aleorithm values had definitely conv®rged(T, . < T„ .
~ 139.9 F
1,J 2, j
after 60 iterations). The conditions under which this algorithm
willconverge to values greater than the exposure surface
temperatures (andhence apparently violate the first law of
thermodynamics) are discussed
2 2in Note 3. This discussion is for T
^ iwhere the
2tion quess” of T. . = 0 i = 2, Xoj j = 1, 2 ...^
j
2 2found that similar unstable conditions (i.e. T. . < T_
)
J • J
2 2if the initial iteration quess was T. , = T. .or even if
"initial itera-
Yq. We also
occurred even
T^ .j were
assigned their exact values as predicted from theory (the
convergence
of T2
2.jwas still 140 °F).
The present status of the project is to circumvent this problem.
Itmay be that the constraints involved in keeping the Crank
-Nicolson algorithmfrom violating the first law^ a At several
orders of magnitude smaller willhave to be used.
-
*> .! frMr* ' ^... , .V (• /r< • t Ai^T
M-J'A
ofr I -
I ,' ,
^
f, t/|r»f:--.'
' I- < »• A'-J . *^;^^.:' VV. -.
i - • •',*- ' U-- :-. ^ *’ '. ,'Z^-' '
,f.4 v.'» 1 t‘ tj .. . ' ,;.
(t ,1 ,y^{i '-if ' '* 41% *J|J»
• -,
#«5'r **» i' «P
" ft‘ '
'_ I ,v,t:.
P t - ^ :[» 5s.->i. ;_—. . -y.h- K -f'^’. 'm
): ••:. Xi
ec '«.•'•
• •“• '^3
-
h
hc
hn
k
]
T
To
Ts
.
0,1
t
y
d
e
P
Ax, Ay
At
NOMENCLATURE*
surface film coefficient
convection heat-transfer film coefficient
radiation heat transfer film coefficient
conductivity
thickness
temperature
unexposed surface temperature
exposed surface temperature
finite-difference notation for temperature at time
increment, x coordinate of iAx = Ax and y coordinate
jAy .. jAx.
t ime
space coordinates
dif fersivity
surface emissivity
density
finite division of x or y coordinate
finite division of time
*Units are English
-
REFERENCES
1. "Principles of Engineering Heat Transfer^" W. Giedt, van
Nostrand Co.,Inc., 1957, p 2.
2. "The Transmission of Heat by Radiation and Convection," E.
Griffiths
& A„ H. Davis, Food Investigation Special Report No. 9,
Dept, ofScientific and Industrial Research, London, England,
1931.
3. "Heat Transfer," M. Jakob, John Wiley & Sons, Inc., Vol
I, 1949,
PP 526-527, 532„
4. "Elements of Heat Transfer and Insulation, " M. Jakob and G.
Hawkins,John Wiley and Sons, Inc., 1950, p 99.
5. "Heat Transfer," Mo Jakob, John Wiley & Sons, Inc. 1957,
Vol II, p 304.
6. "Conduction of Heat in Solids," Carslaw & Jaeger, 2nd
Edition, Oxfordat the Clarendon Press, 1959.
7. "Numerical Methods in Fortran," J. McCormick & M.
Salvador!, Prentice-Hall, Inc., 1964, PP 21, 79, 208-209.
8. "The Numerical Solution of Parabolic and Elliptic
Differential Equations,"D. Peaceman and H. Rachford, Jr.,
Industrial and Applied Mathematics,Vol 3, 1955. PP 28-41.
9. "Introduction to Numerical Analysis," F. Hildebrand.
McGraw-Hill BookCo., Inc., 1956.
10. "Numerical Methods for Scientists and Engineers," R. W.
Hamming, McGraw-Hill Book Co., Inc., 1962.
11. "Elements of Numerical Analysis," P. Henrice, John-Wiley
& Sons, Inc.,1964.
12. "Iterative Methods for the Solution of Equations," J. Traul,
Prentics-Hall, Inc., 1964.
13. "Advanced Calculus," R. Buck, McGraw-Hill Book Co., 1956. p
255.
14. "Tests of the Fire Resistance and Thermal Properties of
Solid ConcreteSlabs and Their Significance, " C. Menzel, ASTM, Vol
43, 1043, pp 1136-1145.
15. "A Practical Method for Numerical Evaluation of Solutions of
PartialDifferential Equations of Heat-Conductive Type," J. Crank
& P. Nicolson,Proc. Comb. Phil. Soc., 43, 1947, pp 50-67.
-
References (continued)
16. "A Stable, Explicit Numerical Solution of the Conduction
Equation forMulti-Dimensional Non-Homogeneous Media, " Allada &
Quon, AIChEPreprint 27, pp 1-16.
17. "A Treatis on Theoretical Fire Endurance Rating," T. Z.
Harmathy,ASTM Special Technical Publication No. 301, 1961, pp .
10-35.
18. "Thermal Radiation Properties Survey." Gubareff, Honeywell
ResearchCenter, Minn-Honeywell Regulator Co., Minn., Minnesota,
1966, p 197.
19. "An Error Analysis of Numerical Solutions of the Transient
Heat Con-Duction Equation," D, B. Mitchell, Thesis, Air Force
Institute ofTechnology Air University, 1969, pp 2-3.
20. "The Effect of Latent Heat on Numerical Solutions of the
Heat FlowEquation," P. Price & M, Slack, Brit. J. Appl. Phy.,
Vol 5, 1954,pp. 285-287.
21. "Heat Conduction in a Melting Solid," H. Landau, Quarterly
of AppliedMathematics, Vol. 8, 1950, pp 81-151.
22. Private conversation with Dr. Robert J. Arms at the NBSo
-
Note 1
A DISCUSSION OF WHEN TO TERMINATE CALCULATIONS
k+1OF T . BY GAUSS-SIEDEL ITERATION*
J
Consider the equations
(4 + ^7^) . + T^^J . + 1 + T^"*"! . + .kAt 1,.] 1-1,1 i+l,J
i,J+l J-1 i,j ( 1 )
+ Ax ^ + 2) T^"-^ = ^ (T^-"! . + .kAt k i,j 1-1, J 2 i,J-l i,J+l
i,J( 2 )
where a°i,j = T^^ . + t; . + t. + T‘f . , - 4T,' 1-1, J 1+1,
J
^ + T^ - 4T^iJ+1 iJ-1 i,j
a" i, j = TV . . + ^ (T^^ . , + T;' ) + (' 1-1, J 2 i,j-l 1,
J+1
pc .,h
-Ax f -2)T? .kAt k 1, j
Equation 1 is the Crank-Nicolson (CN) algorithm for finding
interiortemperatures of a solid bounded by two infinite parallel
planes. Equa-
tion 2 is the CN algorithm for the unexposed surface
temperatures. (The
solid is NAx thick and MAy tall (Ax = Ay)). Since the equations
arediagonally dominate, Gauss-Siedel Iteration techniques are used
to solvethe N'M unknown „k+l .. m.i , too • , •T^
^
(i = 2,3, ..., N+1; j = 1,2, 3 M; i = 1 is
the unexposed surface).
k k+1All T. , and T. . are known.
i,J
k+1As an initial start to the iteration, all T. , are set equal
to
' 1. J
2 2zero. Equation 1 is used to calculate T„ ^ . This new value
of T„
z, 1 2,12
is used now in the calculation of T„ The iteration process
of
k+1replacing the old values of T^ . by the newly calculated
values of
k+1T. . continues methodically by working through the j = 1,2,
... M for
J
* The method of determining the termination of the
calculationsis mainly from (21) . See also (9)
.
-
Note 1
- 2 -
k+1each i index until a new set of T
, ,have been iterated.
J
Let
n k+l lc+1AT
, , = (T (nth iteration) - T. . (n- 1 iteration)
(In the following discussion^ it is convenient to consider only
valuesof n > 3 . )
For our diagonally dominate matrix. AT? . should become
small
after a finite number of iterations. That is^ we assume
lim at: . = 0J
n-e eo
i = 2,3, . . .,N
j = 1,2,...,M
In practice, this process must be terminated after a
finitenumber of iterations (Iq)* This is called truncating. Let S.
.
J
be the '"exact" temperature value at [(i+l)Ax, JAy]. The radius
of
k+1convergence is given by (T . . in this formula assumes the
value of the
io iteratioii)
:
r.k+1T. . - S. .
J iJ< 1>
1-LAT^ .
^,2
where* L = supn=3
supi =
j = 1, 2. . o,Mi = 2,3. . .,N
jat’^'^^:
1
—I
AT^' J
That is, for the (i+1) iteration, we find the maximum value \t
of the
-T+1 i
ratioATu.
AtJ . 1J
A._ = sup
i = 2,3,
j - 1.2, M
AT+1
AT. .
ATT .J
reference 13, pg 9
-
Note 1
- 3 -
From the set of (3 < T <
-
Note 2
GRID SIZE DETERMINATION
The following houristic method of arriving at an equation
todetermine the proper grid size to use in one-dimensional finite
dif-ference equations was described by Dr. Arms (21). We can extend
theresults to two-dimensional problems. Let f. ((i+1) Ax) be the
itera-tived temperature value for grid size Ax. Call f((i+l) Ax)
the "exact"value. Then^ in an approximate manner^
£Ax ((i+l)^x) = f((i+l) Ax) + Ax^.A
A can be thought of as an operator. (The series expansion was
terminatedon the first term). Taking a grid size of Ax/2
'Ax/
2
= f +Ax^
orf . - f Ax/2Ax
A + 2nd order terms.
By doing the same problem for two different grid sizes^ A can
bedetermined (we neglect the 2^^^ order terms), and hence the
propervariable grid spacing for the desired degree of accuracy.
-
- s*.-
.
-
Note 3
A DISCUSSION OF THE CONSTRAINTS
IMPOSED ON CRANK-NICOLSON FINITE -DIFFERENCE
ALGORITHM BY GAUSS -SIEDEL ITERATION
Consider a solid (i thick) bounded by two parallel planes. Onone
surface^ there is a sudden temperature rise Ts for t>0. The
ini-tial temperature of the solid is zero. Using Crank-Nicolson
differenceequations^ we wish to find the interior temperatures at
time At. Gauss-Siedel Iteration will be used to solve for T^ • . As
a first quess setall T? . equal to zero. The equations for a
irariable Ax and Ay are:
J
i=l
exposedsurface
for t>o)
inside solid:
(9sAll. + ^ + AiL + 1)^kAt Ax^ (Axj^+Ax^) Ax2(Ax^+Ax 2> i,
j
^Ax^(Ax^ + AX2)
k+1 Av^ k+1 1i-l,j Ax2(Axj^ + Ax^) i+1, j 2 J-1
+
Tk+1i, j+1
) ( 1 )
_Ay^ k Ay^ k+1 1 .k+1
^i,j“ Axj^CAy^ + Ax2> Ax2(Ax^ + Ax2> i+1, j 2 ^ i, j-1
)4f£gAyl ^ k
i^ j+2^ \kAt Axj^(Ax^ + Ax^) Ax2(Ax^ + Ax^) / ^i
-
Note 3
- 2 -
at surface 'k = i
i+l,j
Axj k^
kA: ^ ^i,J
/AZ N2 ™k+l'^Ax/
, 1 /„k+l , „k+l . ,i-lj 2 ^i,j+P * \ j
^kAt- (Ax,
Ay hAx k
' 1) T
let Ax-i = Ax, Ay i = 2^ • *1 1
m.
Equation (1) then becomes (noting for our case = 0);
i, j^
"i+1^ ^
^^i,j-l i,j+r
Where 0 = (2pcAy'‘
k At
-1+ 2(gr +2) ^ < 1
Let us introduce some symmetry into the boundry condition:
_k k^i, 1 - ^i,3
Then the first iteration is:
-
Note 3
- 3 -
T? . = gu)T (1 + P) =Zm
y^ 8 ^y J-
T24
= ^“^8 ^
2 Q ^ /I I Q . q 2 . , Qin-2. Q „T» = P(A)T (1 + P + P + .00 + P
) = PodT (t
—
Q )
2, m s s i-p
2 2 2t:^ „ = pVt3, 2 8
„ = pVt (1 + 2 P)3, 3 s
„ = pVt (1 + 2p + 3P^)J j s
T? = P^OD^ n + 2P + 3P^ +3,m s
+ (m-1)l-p”~^-(m-l)p"^
- (1 - P)^
etc
,
second iteration:
- = P [u)T + P^U)\ + 2Puyr (1 + p)12, 2 s s s
= PtuT (+ 2P + 2P^ + P^oo^) < 1s
For the various heat-conclusion problems under study^ this
typeof analysis was used to understand the constraints imposed on
the iterationtechniques. It was found that when the constraints
were violated, therewas convergence of the (all equations had
diagonal dominance), but
J
k+1 kthe
jwere larger than for several Ax^ from the exposed surface
^).
-
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-
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