Transcript Two-Thickness Procedure of FOX50 Tests

Revised Theory
of the Slug Calorimeter Method
for Accurate Thermal Conductivity
and Thermal Diffusivity
Measurements
Akhan Tleoubaev, Andrzej Brzezinski
Presented at the 30th International
Thermal Conductivity Conference and
the 18th International Thermal Expansion Symposium
September, 2009, Pittsburgh, Pennsylvania, USA
Slug Calorimeter
The Slug Calorimeter Method for thermal
conductivity measurements of the fire
resistive materials (FRM) at temperatures
up to 1100K (~827C)
• ASTM E2584-07 “Standard Practice for Thermal
Conductivity of Materials Using a Thermal
Capacitance (Slug) Calorimeter”
Committee
E37 on Thermal Measurements
Approximate formula for thermal
conductivity used in the Slug
Calorimeter Method until now:
 l (T’/t)x(M’Cp’+MCp)/(2AT)
was obtained using the 2nd order polynomial:
T(z,t)  T’(t) + a(t)z + b(t)z2
More accurate approximate formula
for thermal conductivity can be
derived using the 3rd order
polynomial:
T(z,t)  T’(t)+a(t)z+b(t)z2+c(t)z3
Thermal problem:
P.D.E.:
 x 2T/z2 = Cp x T/t
B.C. at z=0: C’p’l’/2 x T’/t=  x T/z
B.C. at z=l:
T(l,t)=Ft
z=0
Coefficients a(t), b(t), and c(t) can be found
as:
a(t)= C’p’l’ x (T’/t)/(2) -from B.C. at z=0
b(t)=Cp x (T’/t)/(2) -from P.D.E at z=0
c(t)=[F- (T’/t)] x Cp/(6l) -from P.D.E.
and B.C. at z=l
New formula for thermal conductivity:
  (l/2)[(T’/t) ( C’p’l’ + Cpl ) +
+ (Cpl/3)/(F- T’/t)] /T
Volumetric Specific Heat Cp
can be found by recording the slug’s T’ relaxation
when the outer T (z=l) is maintained constant.
Same thermal problem:
P.D.E.:
 x 2T/z2 = Cp x T/t
B.C. at z=0: C’p’l’/2 x T’/t=  x T/z
Only B.C. at z=l now is:
T(l,t)=0
z=0
Regular Regime
(A.N.Tikhonov and A.A.Samarskii
“Equations of Mathematical Physics” Dover Publ., 1963, 199
Regular Regime
At large t the sum of the exponents degenerates into a
single exponent:
T(z,t)  exp{-k12t} [A1cos(1l)+B1sin(1l)]=
= exp{-t/} T(z)
This late stage is the so-called “regular regime”
Where
 =1/(k12)
is relaxation time
T(z) is the time-invariant temperature profile
Analytical solution of the thermal problem is
a transcendental equation for eigenvalues
(M. Necati Özisik “Boundary Value Problems of Heat Conduction”
Dover Publications, 1968):
(ml) x tan(ml)/2 = Cpl /(C’p’l’)
where m are roots of the equation.
Relaxation time  can be calculated from the slope
of the logarithm of T vs. time. Reciprocal of the
slope equals
 =1/(k12)
Experimental check of the solution was done using
1/8”-thick copper plate
and two ½”-thick EPS samples.
System of two equations with two
dimensionless unknowns:
• 1) Dimensionless thermal similarity parameter:
 = 1l = l/(k)1/2 = Fo-1/2
• 2) Dimensionless ratio of the specific heats (and
thicknesses):
Cpl /(C’p’l’)
Solution of the system is in another
one transcendental equation:
• f() = (Ft-T’)/(dT’/dt)/ -
- 2[(F/6/(dT’/dt)+1//tan()+1/3] = 0
which can be solved by iterations using e.g.
Newton’s method:
[j+1] = [j] – f ([j])/ f’ ([j])
f’ ()=-F/3/(dT’/dt)-1/tan()+/sin2()-(2/3)
Percent of errors of the slug’s T’ calculated usin
old and new formulas vs. time in seconds.
Thermal conductivity vs. time calculated by old, and by new
formulas using known Cp, and by new formula without
using the known Cp, but using only accurately measured
relaxation time 
Calc-d thermal conductivity, W/mK
0.1800
0.1700
0.1600
Th.Cond.old
0.1500
Th.Cond.new
0.1400
Th.Cond.new 2D
0.1300
0.1200
0.1100
0.1000
0.0900
0.0800
0.0700
0.0600
0
600
1200
1800
2400
3000
3600
4200
4800
5400
Time, seconds
6000
6600
7200
7800
8400
9000
9
Conclusions
• Theory of the Slug Calorimeter Method was revised.
More accurate formula has been derived for thermal
conductivity calculations.
• Volumetric specific heat ratio can be obtained using
another one new formula and accurate registration of
the system’s relaxation time.
• Experimental check proved validity of the new formulas.
Thus, in general, all four thermal properties , Cp, k,
and  can be measured using the new formulas and twostep procedure – first, maintaining the outer
temperature constant, and then changing it at a
constant rate.