Revised Theory Revised Theory

of the Slug Calorimeter Methodof the Slug Calorimeter Method

for Accurate Thermal Conductivity for Accurate Thermal Conductivity

and Thermal Diffusivity and Thermal Diffusivity

MeasurementsMeasurements

Akhan Tleoubaev, Andrzej BrzezinskiAkhan Tleoubaev, Andrzej Brzezinski

Presented at the 30Presented at the 30thth International International

Thermal Conductivity Conference and Thermal Conductivity Conference and

the 18the 18thth International Thermal Expansion Symposium International Thermal Expansion Symposium

September, 2009, Pittsburgh, Pennsylvania, USASeptember, 2009, Pittsburgh, Pennsylvania, USA

Slug Calorimeter Slug Calorimeter

The Slug Calorimeter Method for The Slug Calorimeter Method for thermal conductivity measurements thermal conductivity measurements of the fire resistive materials (FRM) of the fire resistive materials (FRM) at temperatures up to 1100K at temperatures up to 1100K (~827C)(~827C)

• ASTM E2584-07ASTM E2584-07 “Standard Practice for “Standard Practice for Thermal Conductivity of Materials Using a Thermal Conductivity of Materials Using a Thermal Capacitance (Slug) Calorimeter”Thermal Capacitance (Slug) Calorimeter” Committee E37 on Thermal MeasurementsCommittee E37 on Thermal Measurements

Approximate formula for thermal Approximate formula for thermal conductivity used in the Slug conductivity used in the Slug Calorimeter Method until now:Calorimeter Method until now:

l (l (TT’’//t)x(M’Ct)x(M’Cpp’+MC’+MCpp)/(2A)/(2AT)T)

was obtained using the 2was obtained using the 2ndnd order polynomial: order polynomial:

T(z,t) T(z,t) T’(t) + a(t)z + b(t)z T’(t) + a(t)z + b(t)z22

More accurate approximate More accurate approximate formula for thermal conductivity formula for thermal conductivity can be derived using the 3can be derived using the 3rdrd order order polynomial:polynomial:

T(z,t) T(z,t) T’(t)+a(t)z+b(t)z T’(t)+a(t)z+b(t)z22+c(t)z+c(t)z33

ThermalThermal problem:problem:

P.D.E.:P.D.E.: x x 22T/T/zz22 = C = Cpp x x T/T/tt

B.C. at z=0:B.C. at z=0: C’C’pp’l’/2 x ’l’/2 x T’/T’/t= t= x x T/T/z z z=0z=0

B.C. at z=l:B.C. at z=l: T(l,t)=FtT(l,t)=Ft

Coefficients a(t), b(t), and c(t) can be Coefficients a(t), b(t), and c(t) can be found as:found as:

a(t)= C’a(t)= C’pp’l’ x (’l’ x (T’/T’/t)/(2t)/(2)) -from B.C. at -from B.C. at z=0z=0

b(t)=Cb(t)=Cpp x ( x (T’/T’/t)/(2t)/(2)) -from P.D.E at -from P.D.E at z=0z=0

c(t)=[F- (c(t)=[F- (T’/T’/t)] x Ct)] x Cpp/(6/(6l)l) -from P.D.E. -from P.D.E. and B.C. at and B.C. at z=lz=l

New formula for thermal conductivity:New formula for thermal conductivity:

(l/2)[((l/2)[(TT’’//t) ( C’t) ( C’pp’l’ + C’l’ + Cppl )l ) + +

+ (C+ (Cppl/3)/(F- l/3)/(F- TT’’//t)] /t)] /TT

Volumetric Specific Heat Volumetric Specific Heat CCpp

can be found by recording the slug’s can be found by recording the slug’s T’T’ relaxation when the outer relaxation when the outer T (z=l)T (z=l) is maintained is maintained constant.constant.

Same thermalSame thermal problem:problem:

P.D.E.:P.D.E.: x x 22T/T/zz22 = C = Cpp x x T/T/tt

B.C. at z=0:B.C. at z=0: C’C’pp’l’/2 x ’l’/2 x T’/T’/t= t= x x T/T/z z z=0z=0

Only B.C.Only B.C. at z=l at z=l now is: now is: T(l,t)=0 T(l,t)=0

Regular Regime Regular Regime (A.N.Tikhonov and A.A.Samarskii (A.N.Tikhonov and A.A.Samarskii

“Equations of Mathematical Physics” Dover Publ., “Equations of Mathematical Physics” Dover Publ., 1963, 1990)1963, 1990)

Regular RegimeRegular Regime

At large At large tt the sum of the exponents degenerates the sum of the exponents degenerates into a into a singlesingle exponent: exponent:

T(z,t) T(z,t) exp{-k exp{-k1122t} [At} [A11cos(cos(11l)+Bl)+B11sin(sin(11l)]=l)]=

= exp{-t/= exp{-t/} T(z)} T(z)

This late stage is the so-called This late stage is the so-called “regular regime” “regular regime”

Where Where =1/(k =1/(k1122) ) is relaxation timeis relaxation time

T(z)T(z) is the time-invariant temperature profile is the time-invariant temperature profile

Analytical solution of the thermal problem isAnalytical solution of the thermal problem isa transcendental equation for eigenvalues a transcendental equation for eigenvalues

(M. Necati (M. Necati ÖÖzisik zisik “Boundary Value Problems of Heat “Boundary Value Problems of Heat Conduction”Conduction” Dover Publications, 1968): Dover Publications, 1968):

((mml) x tan(l) x tan(mml)/2 = Cl)/2 = Cppl /(C’l /(C’pp’l’)’l’)

where where m m are roots of the equationare roots of the equation..RRelaxation time elaxation time can be calculated from the can be calculated from the

slope of the logarithm of slope of the logarithm of TT vs. time. vs. time. Reciprocal of the slope equalsReciprocal of the slope equals

=1/(k =1/(k1122))

Experimental check of the solution was done Experimental check of the solution was done using 1/8”-thick copper plateusing 1/8”-thick copper plate

and two ½”-thick EPS samples.and two ½”-thick EPS samples.

System of two equations with two System of two equations with two dimensionless unknowns: dimensionless unknowns:

• 1)1) Dimensionless thermal similarity parameter:Dimensionless thermal similarity parameter:

= = 11l = l/(kl = l/(k))1/21/2 = Fo= Fo-1/2-1/2

• 2)2) Dimensionless ratio of the specific heats Dimensionless ratio of the specific heats (and (and thicknesses):thicknesses):

CCppl /(C’l /(C’pp’l’)’l’)

Solution of the system is in Solution of the system is in another one transcendental another one transcendental equation:equation:• f(f() = (Ft-T’)/(dT’/dt)/) = (Ft-T’)/(dT’/dt)/ - -

- - 22[(F/6/(dT’/dt)+1/[(F/6/(dT’/dt)+1//tan(/tan()+1/3] = 0)+1/3] = 0

which can be solved by iterations using which can be solved by iterations using e.g. Newton’s method:e.g. Newton’s method:

[j+1][j+1] = = [j][j] – f ( – f ([j][j])/ f’ ()/ f’ ([j][j]))

f’ (f’ ()=-F)=-F/3/(dT’/dt)-1/tan(/3/(dT’/dt)-1/tan()+)+/sin/sin22(()-(2/3))-(2/3)

Percent of errors of the slug’s T’ calculated Percent of errors of the slug’s T’ calculated using old and new formulas vs. time in using old and new formulas vs. time in seconds. seconds.

Thermal conductivity vs. time calculated by old, and Thermal conductivity vs. time calculated by old, and by new formulas using known by new formulas using known CCpp, and by new , and by new formula without using the known formula without using the known CCpp, but using only , but using only accurately measured relaxation time accurately measured relaxation time

0.0600

0.0700

0.0800

0.0900

0.1000

0.1100

0.1200

0.1300

0.1400

0.1500

0.1600

0.1700

0.1800

0 600 1200 1800 2400 3000 3600 4200 4800 5400 6000 6600 7200 7800 8400 9000 9600

Time, seconds

Ca

lc-d

th

erm

al

co

nd

uc

tiv

ity

, W

/mK

Th.Cond.old

Th.Cond.new

Th.Cond.new 2D

ConclusionsConclusions• 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 two-step procedure – first, maintaining the outer temperature constant, and then changing it at a constant rate.