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Thermal properties of concrete at elevated temperatures

Harmathy, T. Z.

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T . 2. Harmathy'

Thermal Properties of Concrete

at Elevated Temperatures

REFERENCE: Harmathy, T. Z., "Thermal Properties of Concrete at Elevated Temperatures," l o ~ l r r ~ a l of Mnterials, JMLSA, Vol. 5 , No. 1, March 1970, pp. 47-74.

ABSTRACT: The practice of regarding the thermal properties of con- crete as constants may lead to serious errors in heat flow calcu1a:ions. Because of the physicochemical instability of concrete it is very difficult to procure valid experimental information. [n this paper theoreiical con- siderations are presented which, together with some experimental data, will make the assessment of the thermal properties of concrete possible. The calculation procedure is illustrated by examples.

KEY WORDS: concrete, portland cement, concrete aggregates, light- weight aggregates, elevated temperatures, thermal conductivity, specific heat, bulk density, true density, tobermorite gel, dilatometry, thermo- gravimetry, porosity, enthalpy, dehydration reactions

Nomenclature

Specific heat, cal/g deg K Heat capacity cal/mole deg K Group heat capacity, cal/mole deg K True density (when d # ,D), g/cm3

Coefficient of diffusion of water vapor in air, cm2/s Emissivity, dimensionless

Enthalpy, cal/mole Group enthalpy, cal/mole Heat of reaction, cal/mole

Thermal conductivity, cal/cm s K

Fictitious thermal conductivity, cal/cm s K Length of dilatometric specimen, cm Change of length, cm

Characteristic pore dimension, cm Maturity factor, dimensionless

'Fire Section, Division of Building Research, National Research Council of Canada, Ottawa, Canada.

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4 8 JCURNAL OF MATERIALS

Molecular weight, g/mole Group molccular weight, g/mole

Constant of phase distribution geometry, dinlensionless Equilibrium vapor pressure of adsorbed moisture, g/cm s2 Equilibrium vapor pressure of free water, g/cm s'

Total pressure, g / c n ~ s' Heat of sorption, cal/mole Gas constant, cal/mole deg K

Temperature, deg K (unless otherwise stated) Volume fraction, cin:{/cm:'

Weight fraction, g/g Weight, g

Mole fraction, moles/mole Group molc fraction, moles/mole Greelc Letters

Expressions explained by Eqs 26, 28, and 30, dimensionless Factor, defined by E q 42, dimensionless

Porosity, cm3/cm3

Heat of vaporization, cal/mole

Reaction progress variable, dimensionless Bulk density or true density if p = d, g/cm3 Stefan-Boltzmann constant, cal/cm2 s K.'

Stoichiometric constant, dimensionless

Volumetric fraction of moisture (evaporable water), cmVcm3 Subscripts (Component Nutnbers a n d Clzetnical Fornzulas Not Included)

Of aggregate

Of air, of air in air cavities Of cement paste

Of cement in its anhydrous state Of concrete

Of the i-th component Normal to plane of slabs Of nonevaporable water Of water in fresh paste Parallel to plane of slabs At constant pressure Due to radiant-heat transfer Of shale

Of toberrnorite gel A t constant temperature Of water

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H A R M A T H Y O N CONCRETE 4 9

'P Due to moisture transfer

298 At T = 298 dcg K

1273 At T = 1273 deg K

Before the advent of computers, the calculation of heat flow in con- crete constructions during fire exposure or in the concrete radiation shield of nuclear power stations was not possible without a number of simplifying assunlptions concerning the heat flow process, the geometry of the construction, and the material properties. Today most of these assumptions are no longer necessary. O n the contrary, inore ancl more sophisticated numerical ~llethocls have been developed which take into account to the last detail the true nlechanisnl of heat flow and the complex geometry of the construction. Yet even the nlost so- phisticated nunlcrical procedure will yield grossly inaccurate results if it is based on unreliable i n p ~ ~ t information concerning the material properties.

A n increasing need for such information in numerical fire endurance studies prompted several years ago a comprehensive research into the thermal properties of concrete at elevated temperatures. The Boulder Canyon Report [ I ] ' was the first publication to call attention to the fact that (depending on the mix proportions and the aggregates used) various concretes may exhibit widely different thermal properties even at room temperature. At higher temperatures the developing decomposi- tion ancl transition reactions add to the coinplexity of estimating the thermal properties of concrete. Yet, with the aid of some simplifying assumptions, it is possible to make such estimations with reasonable accuracy.

The application of the information to be presented in this paper will be illustrated by re-examining the thcrinal properties of four previ-

ously described

1-31

hypothetical concretes in the 25 to 1000 C tempera-

ture range.

Stoichiometric Modeling of Cement Paste

Typical portland celllent clinkers contain 4 5 percent of tricalcium silicate (C:,S),:' 25 percent of 8-dicalcium silicate (P-C,S), some C-:A, C.,AF, calcium sulfate, and a sm;~ll amount of other compounds. The

mixture of con~pounds forn~ecl by hydration from the cement is called,

at any stage of hydration, cement paste. A major part of the paste

consists of an impure calcium silicate hydrate of somewhat indefinite composition that forins froin the C::S and P-C,S components of the

cement 13-61. This hydrate is commonly referred to as tobermorite

' T h e italic numbers in brackets reEer to the references at the end of thc paper.

T is an abbreviation for CnO. S for SiO?, A for Al10.:. F for Fe,O.:, and H for H1O. Thus C.:S means 3CaOaSi0,. etc.

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50 JOURNAL OF MATERIALS

gel. In addition to it the paste contains a large number of less easily identifiable colloidal hydrates formed from other compounds of the cement, some fairly well developed C H crystals, and some anhydrous residues of the cenient.

The amount of water that the paste holds by chenlical bonds (non- evaporable water) cannot be unambiguously separated from the amount held by adsorption (evaporable water). The former is defined as the water retained by the paste after it has been dried either by the dry ice method [7] or in a furnace of 105 C.

The nonevaporable water content of a niaturc paste can be estimated from the compounci composition of the cement with the following for- mula [ 8 ] :

Most actual cement pastes are not completely mature and contain a small amount of the anhydrous cement. Consequently the value of

w,,/w, is less than that calculable from E q 1.

T o avoid some difficulties created by the presence of a number of poorly identifiable A- and F-containing hydrates, it seems logical to idealize the paste as if it were formed from a cement containing C , S and 8-C.S only.

In Table 1 the compound composition of a portland cement is shown. This cement was used in various pastes, some properties of which were examined in the author's laboratory. The idealized composition of the cement, calculated by retaining the actual C S to p-C,S ratio, is also shown in the table.

According to Eq 1 this cement yields a paste for which, at com- plete ~iiat~irity, w,,/w, = 0.21 6. The amount of water to be combined

~\c.tunl Itlc:~lizc~l Corr~l,ositiorl C k ~ r l ~ p o u r ~ t l C o r ~ ~ p o s i t i o l l , -- -- -

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H A R M A T H Y O N CONCRETE 5 1 with one mole of the idealized cement to yield the same w,,/w,:: ratio is

where the information given in Table 1 was utilized.

From the work of Brunauer and Greenberg

[4]

it appears that the average composition of the tobermorite gel may be described by the formula C,,,;,SH,,,. The idealized hydration scheme for the cement of Table 1 thus may be written as

Here the maturity factor, nz, takes care of the fact that the reaction may not have developed to completion. Based on information pre- sented by Taplin [9] .and Powers [ l o ] , 112 is plotted as a function of W , / W ~ : ~ ; and the age in Ref I I.

Three cement pastes, made from the cement described in Table 1 and referred to hereafter as Pastes 1 , 2, and 3, were subjected to various experimental studies. They were prepared at three different water-cement ratios, rv,/w,.:,, = 0.25, 0.33, and 0.5, respectively. Only Paste 3 will be discussed here in detail. The idealized composition of this paste, as well as that of the completely mature paste (in = 1.0) are given in Table 2.

Upon heating, the dehydration of the paste starts as soon as (or even before) the desorption of evaporable water is completed, and

T A B L E 2-ldea11:ed c o i ~ p o s i l i o n o j 11r.o ce~jienl pastes eel ~.oonc letilprrc~/lcre ant1 at 1000 C .

Colnposition, fraction

-

-Pnstc 3,

Tclnpcrnturc Coniponcnt u*,,/w,* = 0.30 1Int11l.c Pnstc, I ) L = 0.S24 111 = 1 .OOO C1.6.'SIll.5. . . . 0.610 O.46:3 0 . 7 1 7 O . > l : j C I I . . . . . 0.241 0.438 0.2s:; 0.4S7 2 5 C C:jS.. . . 0.095 0.056 . . . . . . P-C?S. . . 0.054 0 . 0 4 3 . . . 1 ,000 1 . 000 1 . 000 I . 000 To bcr~noritc gcl gloup 0 . 610 0 . 4 6 3 0 7 0.513 CII g r o u p . . . . . 0.214 0 . 4 1 0.4S7 1000 C CjS. . . 0.111 0.03G . . . . O-C2S.. . . 0 . 0 6 5 0.04:3 . . 1.000 1 ,000 1.000 1.000

xo,rr:.-~Iolcculnr weiglits: C I . G ? S I I I . ~ = 17S.0 g/lnolc CII = 74.1 g/molc CS = 116.1 g/molc

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5 2 JOURNAL OF MATERIALS

proceeds uninterruptedly up to temperatures in excess of 800 C. The final dehydration residue of tobern~oritc gel consists mainly of p-wol- lastonite (p-CS) and P-CIS. The C H reduces into C. In the present idealized treatment the following assumptions will be used: ( a ) There arc only two dehydration processes, one concerning the tobermorite gcl and one the C H ; ( b ) as the dehydration proceeds, it is invariably the final dehydration products that form; in other words, there are no intermediate dehydration products; ( c ) the progress variables for the reactions are unique functions of the tcmperaturc. Accordingly, the dehydration at any temperature is described by the following two schemes :

F o r convenience the solid mixtures represented by the bracketed terms in Eqs 3 and 4 will be treated as single compounds and will be referred to as the "tobcrmorite gcl group" and the "CH group," respectively. Naturally, these fictitious conlpounds have variable nlolecular weights,

and

-1Iclr = Jlc:~, - ,tcl~;ll~l. . . . . . . . ((i) The completion of the dehydration reactions is represented by the

Temperature C

FIG, 1-Dcgree alld mite of co~~versiorz of to6r1.1norite gel i r ~ 0 1 1 idacilized cc1~1e11t pcrste.

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HARMATHY ON CONCRETE 5 3

;,

= 1 and

<,.,,

= 1 conclitions. Tablc 2 also contains information on

the composition of thc previously dcscribecl paste at 1000 C wherc both dehydration reactions are known to havc dcvclopccl to completion.

The c I ( T ) and & , , ( T ) functions have been estimated with the aid of thermogravimctry and diffcrcntial thermal analysis and are plotted in Figs. 1 and 2.

Temperature C

FIG. 2-Degree crt~d rnrc of L . o I ~ ~ ~ ~ ~ : Y ~ o I I of C ( I / L . ~ I ~ I ~ I /~)ldro.vidc, it1 (111 it1enli:rd

cerrierlr paste. Dejir.ee crrltl rntr of ~ro~tsforrrlntior~ o f ( l ~ ~ a r t : .

I n a general sense the desorption of evaporable water that tnkcs place around 100 C can also be treated as a reaction. T h e scheme of this "reaction" is

where again < , , ( T ) may be idealized as being a unique relationship. A n estimated form of this relation is given in Fig. 3.

Some Basic Considerations

Additivity Tlzeoret?~

T h e additivity theorem provides simple ways of calculating the den- sity, porosity, and specific heat of multicomponent solid systems. T o the calculation of bulk density thc following for111 of the theorem is applicable :

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5 4 J O U R N A L O F MATERIALS

i ' + c t.ruz dcnsity of n solicl c;ui bc calculated analogously. For exam-

pic:, when ihe solid is spccificcl with thc wcight fractions of its com- iw:-icnts, its truc dcnsily c;in bc cstimatecl as follows:

1r.l:. :-: tllc

7

w ,

-

1 relation was utilized.

L.,

- 7 ,

i

i he porosity can ~ h c n bc obtained either- with ~ h c aid of t.hc familiar f<>~i111.11a

1 3 - r'i 1 1 1 1 thc ~ ' o r o s i t ~ . of 1,llc c r ~ n ~ l ~ o ~ l c r ~ t s , as

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HARMATHY O N CONCRETE 5 5

such as concrete and most coarse-grained stones. In the case of soiids of strongly amorphous structure or containing some of their compon,:iils in solid solutions, Eq 10 can only be regarded as a crude approximarion.

The finding that the heat capacity of solid compounds is approxi- mately equal to the sum of the heat capacities of the component elements is referred to as the Neurnann-Kopp law. This law can further be ex- tended to the calculation of the heat capacity of glasses by using einpiri- cal heat capacity increments for the oxide constituents [ 1 2 ] :

The author found that this additivity theorem was also applicilt)le to the calculation of the heat capacity of calcium silicate hydrates, on which very little information is available. Experimental data 1133

indicated that the heat capacities of these compounds can be predicrzd at a fair accuracy by assigning the following empirical increments to the constituent oxides:

for C,

c;

= 11.67

+

1.08 X 10-Y1' - 1.56 X 1OZ7'-? '

for S, C; = 11.22

+

8.20 X 1 0 - T - 2.70 X 1O5T-'

I

(IS) for H ,

c;

= 6.49

+

G.GO X lOWT - 0.91 X 105T-?

Thus, for the tobermorite gel (C1.ceSHl.j) O I ~ C obt:liils

(CrJ)t = 39.SG

+

19.S.5 X 1 0 - T - 6.55 X 10j7'-? . . . . ( 1 (j)

Naturally, above 100 C this equation applies only to the undecomposed part of the gel.

I n the case of multiphase solid mixtures, Eqs 13 and 14 are strictly valid.

Heat Capacity o j Unstable Conlpo~inds

Owing to certain experimental difficulties, it seems unavoidable for the time being to develop the heat capacity versus temperature relation for many chemically unstable compounds on a semitheoretical ground. The heat capacity is defined as

When a compound undergoes a chemical reaction for which the reac- tion progress variable,

t,

is a function of temperature, at any given temperature for which 0

<

,t

<

1 there will be generally a "group" of

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5 6 JOURNAL OF MATERIALS

conlpounds prescnt consisting of the original compound and the re- action products. T h e enthalpy for this group, H, is a function of P, T, and [, so that at P = constant

Thus, bj. virtuc of

Eels

17 t o 20

where

cIJ

is the "group hcat capacity" and

Cp,[

is the "group sensible heat capacity," that is, thc heat capacity in the absence of chemical reaction. ( I t is usually this CP.$ that can be determined by drop c a l o r i ~ ~ ~ e t r y . ) The second term on the right side of E q 2 1 represents the heat capacity associated with thc absorption of heat in the reaction and may bc referred to as the "group latent heat capacity." In this term

where the ( H r , , , ) , is to be referred to a conveniently selected level. I n the decomposition reactions of cemcnt pastc the various oxides are the simplest compounds; it is convenient, therefore, to choose the zero level of enthalpy as the enthalpy of oxides at 2 5 C. With this selection the enthalpy of more co~nplex compounds at 25 C becomes equal to their standard heats of formation from oxides.

Thermal Condrlctivity of Mrlltiphnse Solicl Mixtz1r.e~

T h e additivity theorem is applicable to the calculation of thermal conductivity of multiphase solid mixtures only when the phases are arranged in layers and the heat flow is parallel to the plane of the slabs. If this is the case,

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HARMATHY O N CONCRETE 5 7

is 110 101lger vitlicl, but

A simple formulit nl)plic:lble to :~11y t\\-o-ph;~sc solid misture is as

~ O I I O \ \ - S [14] :

u 1 i i l

+

otu21i2

-

-u1

+

mu?

where

Here Phase 1 must always be the principal continuous phase. n is a function of the phase distribution geometry and, in general, has to be cleterillined experimentally. For 17 --t v j and n = 1, E q 2 5 converts

into Eqs 2 3 and 24, respectively. Furthermore, with tz = 2 and 3 the

Maxwell-Eucken relations for a two-phase system are obtained, with the dispersed phase consisting of cylindrical and spherical particles, respectively [15,16]. There is substantial evidence [14,17] that the ap- plicability of n = 3 is not restricted to spherical particles only, but can be extended to any system comprising a continuous and a discon- tinuous phase.

When both phases are essentially continuous, as in the case of ordi- nary porous solids (with air as the second phase), rathcr low values of n seem to be applicable. From a survey of many experimental data available from the literature, n = 1.5 seemed to be a reasonably good average, at least within the porosity range of interest.

By repeated application, E q 25 may be extended to systems consist- ing of any number of phases. In the case of a system comprising a continuous phase (Phase 1) and two discontinuous phases (Phases 2 and 3 ) one can write

n and N are characteristic of the distribution geometry for Phase 2 in Phase 1, and for Phase 3 in Phase 1-2, respectively.

I n the case of an ordinary porous system there are three phases present: the porous solid (Phase 1 ); the moisture, either adsorbed

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58 J O U R N A L O F MATERIALS

to the pore surfaces or as capillary water (Phase 2 ) ; and the air in the pores (Phase 3 ) . These three phases are all essentially continuous. By first regarding Phases 2 and 3 as a separate system, then considering the mixture to consist of Phase 1 and Phase 2-3, one finally obtains

I\-hcre is obtnincd according to Eqs 2.1, :xnd 26 by changiilg thc iiltlcscs from 1 and 2, to :! arid 3, nild

n and N again represent the distribution geonletries for Phase 3 in Phase 2, and for Phase 2-3 in Phase 1, respectively.

When using these equations for the calculation of the thermal conduc- tivity of a porous system, the thermal conductivity of air has to be interpreted as follows:

The k,, term represents the radiant-heat transfer through the pores [18]. Icy is the equivalent thermal conductivity of moisture transfer [19]. In calculations involving variable temperatures the following equivalent form of Eq 33 is more convenient:

In the above equations, Q and the equilibrium relative humidity at 25 C, that is, (pip,,.),,,,, are functions of the moisture content and can be determined experimentally, or estimated as described in Ref I I .

Density, Porosity

Cen~ent Paste

The true density of the cement paste can be calculated by the follow- ing formula [ I 01 :

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HARMATHY O N CONCRETE 5 9

estimated with the aid of Ref I I , and (1, 2 3.15 g/cm', p,, .= 1.22 g/

cm'. The factor 6 has been introduced to take care of the decrease of nonevaporable water content at elevated temperatures:

and WT/W,!,, is obtained by thernlogravimetry. According to these equations (d,),,;, = d,.::, that is, the true density of the paste at 1000 C is equal to that of the initial anhydrous cement (approxi- mately 3.15 g/cm"). This result is, of course, not strictly correct.

The bulk density of the paste can be expressed as 1

+

)I[ (,LL>?~/ IL>,.*)

.

W T / bfr?98

( P C ) =

i

+

P o1 1 ) ( I

+

a ~ / i ? , , ) ~

. . . (37)

where ~ l / l , , , is to be obtained by dilatometry.

The variation of the porosity with temperature can be calculated now by combining Eqs 1 I , 35, and 37.

In Fig. 4 the thern~ogravimetric and dilatometric curves of Paste 2 are shown. Similar curves and Eqs 11, 35, and 37 were used in plotting the variation of c,, d,, and pc with the temperature in Fig. 5 for Paste 3.

Aggregates

The density of many natural rocks is listed in several handbooks. Reference 11 also gives a short list. When no infornlation is available,

FIG. 4-Dila!oruerric nt~rl rl1er.1~7ogrnviti~eIric wlr.ves of Ccnlet~r Pas!e 2. Rnre o f heatirtg: 5 C/rnirl.

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6 0 J O U R N A L OF MATERIALS

2.5 I I I I I I I I I 10.30

0 100 200 300 400 500 600 700 800 900 1000

Temperature C

FIG. 5-Crrlcrtltrtrrl I'LI/IIC.Y of tile t1.1ir ~ C I I S ; ~ ~ , brrlk ~ l o ~ . r i t y , rrr~rl porosity of Cet17orl P ~ l s t e 3.

it is possible to usc = cl

--

2.65 g/crn5, which is an average applicable to the most common rocks. Altcmativcly, (1 may bc cstimatcd from the mineralogical composition (sce Eq 1 0 ) .

In the case of lightweight aggregates, (1 may be estimated or ~ncasured as clcscribcd in ASTM Test for True Specific Gravity of Refractory Matcrials ( C 135-66). T o determine for such aggregates, some special techniques may be needed, such as that dcscribcd by Shidclcr [20],

or one may attempt to apply somc standard methods dcvelopcd for refractory materials.'

As concrcte aggregates arc gcncrally no1 liablc to undergo tlecomposi- tion reactions, thc change in their densities is usually attributable to volume changcs clue to thermal expansion and possibly to some solid transformations. and can often bc disregarded.'

Some information on thc compositions and densities of four aggre- gates (two normal weight and two lightwcight) to be used in the cxam- ples is given in Table 3.

" A S T M Test for Apparent Porosity, Water Absorption, Apparent Specific

Gravity, :und Bulk Density of Burned liefractory Brick ( C 20-46): ASTM Tesl f o r Bulk Density of Granular R e f r a c ~ o r y M:lterials ( C 357-58(1967)); ASTPVI Test for Bulk Density and Porosity of Granular Refractory Materials by Mercl~ry Displacement ( C 493-64).

' U n f o r t ~ ~ n a t e l y , recent thermogravimetric studies indicated that many so-c;illed "c;llcareo~~s" (and alro some "siliceous") fine apgregates, ~rsed especially in norni;ll weight concrete masonry units, undergo substantial decon~position upon heating.

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H A R M A T H Y O N CONCRETE 61

Cor~lpositiorl, \\ ciglit frartior~

- -1 5 ~ p a n d t d Es1,nndcd Quartz Ar~orthositc S l ~ a l t :I Shnlc 13 Co~lstitucrit p = (1 = 2.6.72 p = t l = 2.704 p = 1.-1.iO p = 1.1.30 E = 0 e = 0 t l = 2.!):$0 t l = 2.930 E = O..iO.j 6 = 0.60s . . . SiO? (cluartz). . . . SiO? (glassy). . . . Na?O.Al,O,.GSiO, . . . CaO.A12O3.2Si0? A1203.. . . F c 2 0 3 . . . . CaO . . . Concrete

With the aid of the additivity theorem thc bulk dcnsity of concrete is expressed as

where the v . . , ~ , ~ term (mass concentration of air in air cavities) is negligibly small.

The overall porosity of the concrete, that is, the space per unit vol- ume occupied by air, can be calculated as

i

Four kinds of concrctes will be discussed as examples. These were conceived to represent limiting cases with regard to their thermal prop- erties, thcrefore were chosen to contain one kind of aggregate only: either the poorest or the best in the normal weight and lightweight groups. The composition of these concretes is listed in Table 4. Their bulk densities in oven-dry condition ( v = 0 ) , as calculated with Eq 38, are also given there.

Since most concrete mixes are made with a water-cement ratio of 0.5 or higher, Paste 3 has been selected in Table 4 to represent the cement paste.

In the calculations concerning the thermal conductivity of concrete (to be discussed later) it was assumed that as long as (p

1

0.8 v,~,,

practically all moisture was held in the paste. In the case of lightweight concretes sonle additional watcr may be present in the pores of the aggregates, but only in a 100 percent relative humidity environment.

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6 2 JOURNAL O F MATERIALS

TJZUI,I~; 4-Sonrc injormcllion concerrtiny Co1trrclcs 1 lo /, (in oucn-dry condilior~). Co~nl)ositiorl, fraction

Conrrctc I Concrete 2 Conc~.ctc 3 Concrete 4

p = 2..")3 p = 2.:333 p = 1.417 p = 1.173

\ r t . vol. \vt. vol. tvt. vol. \vt. vol.

T o estimate the bulk density and porosity of concrete at elevated temperatures one may generally use E q s 38 and 39, respectively, and the assumptions that the volu~lle fractions of the constituents and the densities and porosities of the aggregates remain unchanged o n heating.

Specific Heat

Cernent Paste

I n applying E q 21 and the additivity theorem to the two main constit- uents of the idealized cement paste, namely to the "tobermorite gel group" and to the "CH group" the following two equations are obtained:

These equations have been based on the idealized dehydration schemes presented by Eqs 3 and 4. Since the water formed in the reaction leaves the system in gaseous form, its contribution to the "group sensible heat capacity" was omitted from E q s 40 and 41.

T h e "group latent heat capacity" can be calculated with the aid of Fig. 6 which shows the H ( T ) functions for all reactants and products that appear in the dehydration schemes of Eqs 3 and 4. For reasons mentioned in connection with E q 22, the enthalpy of the constituent oxides at 25 C was selected as the zero level of enthalpy. Quartz was

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H A R M A T H Y O N CONCRETE 63

-40 I 1 I I 1 I I 1 I

0 100 ZOO 300 400 500 600 700 800 900 I000

Temperature C

FIG. 6-El~thnlpy of t'nriorls co~r.slitr~errt.s of nrl irlecrlizetl cc,~,~c.~rl p(1str.

chosen as the reference oxide for S, and liquid water for H. Most of the curves were plotted on the basis of heat capacity and

eat

of formation data collected by Eitel 1121. The 25 to 675 C section of the curve for p-C,S was based on estimated values, and is probably accurate enough for engineering calculations.

The course of the curve for tobermorite gel ( C , ,,ISH, , ) above 25 C was calculated with the aid of E q 16. The heat of formation from its oxide constituents at 25 C was estimated from data reported by Learch and Bogue and by Woods, Steinour, and Starke (see Ref 2 1 ) , concerning the heats of hydration of C , S and ,&CIS, and was taken as -35, 140 cal/mole.

(21)

6 4 JOURNAL OF MATERIALS

and 6 wcre used to calculate the heat capacity versus temperature rela- tions for all four components (see Table 2 ) of the idealized cement paste. The results of thcse calculations arc plotted in Fig. 7.

T o obtain the specific heat of the cement paste the additivity theorem can again be used, now assuming the following forin:

where, as pointed out earlier, the "group molecular weights" M , and M,,, are variable quantities (see Eqs 5 and 6 ) . With thc aid of Figs. 1, 2, and 7, and of information concerning the co~upositions (for

I ,

Z ~ O 3b0 460 5 0 0 600 1 0 0 800 900 1000 Temperature C

(22)

HARMATHY O N CONCRETE 6 5

Paste 3

Temperature C

FIG. 8-Specific hcnt of Ceri~elrt Prrstes 1, 2 , c11rcl 3.

Paste 3 see Table 2 ) , the specific heats of Pastes 1, 2, and 3 were cal- culated and are plotted in Fig. 8. It is secn that, owing to the ab- sorption of heat in the dehydration reactions, the cffcctive value of thc specific hcat may be scvcral times highcr than the "scnsible hcat capacity" which is approxi~natcly thc valuc ~ncasurable by drop calorimetry.

Aggregates

The gencral procedure of estimating thc specific heat of concrete aggregates will be illustrated hcrc by examining the four aggrcgatcs specificd in Table 3.

By virtue of E q 17, the heat capacity of quartz can be obtaincd from its enthalpy curvc (Fig. 6 ) by graphical differentiation. At finite rates of heating the

0-p

quartz transformation does not occur instan- taneously (at about 575 C ) but extends over a larger temperature interval. Within this intcrval the hcat capacity of the "quartz group"

(consisting of a and

p

quartz) can be calculated, according to E q 21, as

For simplicity, it was assumed that the & ( T ) function can bc ob- taincd from the <,.,,(T) function by an 80 C clisplaccrnent of thc tcm- perature axis, as shown in Fig. 2.

(23)

66 JOURNAL O'F MATERIALS I I I I I I I I I - - Shales A and B - - - - - - 0 1 - _ 1 _ ' - - i 1 ~ I

d

0 100 200 300 400 500 600 700 800 900 1000 Temperature C

The results of the calculatiorfs concerning the specific hcat of quartz as well as those conccrning the other aggregates of Table 3, arc plotted in F i g 9. Both minerals (albite and anorthitc) that form the anorthosite

arc stable in the 25 to 1000 C interval; therefore the specific heat

versus temperature relation for anorthosite yields a smooth curve. It was calculated with the aid of the additivity theorem and data collected by Eitel [ 1 2 ] .

The expanded shale aggregates are also stable within the 25 to 1000

C interval. The curve for the two expanded shales, as shown in Fig.

9, was calculated from their oxide composition (regarding SiO, as

glassy), using the additivity theorem. Concrete

By making use again of the additivity theorem the following equation is obtained for the specific heat of concrete:

where the last two terms represent, according to the scheme of Eq

(24)

H A R M A T H Y O N CONCRETE 6 7 prescncc of moisture. Regarding moisture simply as frec water in thcsc kinds of calculations is generally pern~~ssiblc, according to the findings of Powers [ l o ] and Berezin et al [22]. At higher moisture contcnts the Q = A approximation is also permissible.

T o avoid the difficulty associatcd with thc fact that pc and the w's in Eq 44 are functions of the temperature, it is advisable to calculate, instead of ( c , ) , . , the "volume specific heat," that is, thc product

( c , ) , ~ , , which in hcat flow analyses is of principal interest. For the volun~e specific heat

where, as mcntioncd enrlicr, thc 1,'s and usually the p,'s can be trcnted

The volume spccific hcats for C'oncretcs I to 4 for = U (oven-dry condition), arc shown in Fig. 10. (Thc curvcs corrcsponding to

= 9.04 and 0.08 have been given in Rcf 2.) The most conspicuous peaks of thcsc curves arc associutcd with the dehydration of cnlciu~n- hydroxide.

One Inay cstin~atc the hcat capacity that would arisc in the absence

1

C o n c r e t e I,

01 I I 1 I 1 I I I 1

0 100 200 300 400 500 600 700 BOO 900 1000

Temperature C

F I G . 10-Calc~rlnlcd vcrlrrcs of Ilze volrrci~e specific lleofs of Coticrefes I to 4 ici o~~cii-rl/.y cot~clitioti.

(25)

68 JOURNAL OF MATERIALS

of chcmical reactions by connecting thc low and high tempcrature sec- tions of thc curves (for w = 0 ) with a straight linc. As mentioned

earlier, it is approximately this value that is measurable with drop calorimetry. It is seen that in the case of ordinary (nonautoclaved) concretes the latent heat contribution to the specific heat is quite sub- stantial. Heat flow calculations based on specific heat values obtained by drop calorinletry are, therefore, liable to erroneous conclusions.

Thermal Conductivity

Cetrlent Paste

Because of the low crystalline ordcr in the toberrnorite gel, the ther- mal conductivity of the cemcnt paste is very low and comparable with that of true glasses. According to the phonon theory of lattice conduc- tion [I61 it is cxpected to be practically independent of the chemical composition (at least at room temperature), consequently also of the age, and to vary in proportion to the heat capacity at elevated temperatures.

By virtue of these a priori concl~isions, one is obviously entitlcd to generalize by saying that the thermal conductivity of any dry cclnent pastc, when cxtrapolatcd to zcro porosity, can be expected to fall in the vicinity of a unique lc: versus T curve.

Ll I I I I I I I I 0 100 200 300 400 500 600 700 800 900 1000 Temperature C 0.004L,, 0.003 U w m E

FIG. 1 I-Thert)lul cotldllcli~liry oj Cetilet~t Pasles 1, 2 , ~ t l d 3.

/

/-.

/ \

Ak;

'

\ \ Exper~mentoi Values

/' \ -

.

\ \ \ \ -

'.

..

---

-

---__

P a s t e I 2

(26)

H A R M A T H Y O N C O N C R E T E 6 9

In Fig. 1 1 somc cxpcriniental valucs conccrning the thermal conduc-

tivities of Pastes 1 , 2, and 3 in thc 0 to 1000 C tcmpcraturc rangc

are shown. Thcsc valucs wcre obtaincd by a variable statc method

described in dctail elscwhcrc (231, and are in substantial agreement

with valucs reported by othcr rcscarclicrs [ 2 4 ] .

As discussed carlicr, n

-

1.5 whcn Eqs 25 and 26 arc applicd to a porous solid. Thus, nftcr combining these two cquations, thc thermal conductivity of cemcnt paste can bc expressed as

After somc rearrangcmcnt this c q ~ ~ n t i o n may also bc ~ ~ s c d for the cal-

culation of the supposedly uniquc Ic: versus T function from experi-

mentally determincd valucs of

X.

and calculatcd valucs of 6 , .

The k,T vcrsus 7' relation, as plottccl in thc clashcd linc in Fig 11, has becn obtained by making usc of thc cxpcrimcntal rcsults conccrning

the thermal conductivity of Paste 2 and of the E , versus T curvc for

this paste. Bcing conccrncd only with the concluctivity of ovcn-dry paste, k , = 0 in the cxpression of I, f (Eq 3 1 ) . Furthermore, owing to the very small valucs of L for ccmcnt pastcs, k,, also turned out to bc a negligibly small fraction of k < even at 1000 C; thus k = k , . Information conccrning thc dcpendence of k , on thc tcmperature is

available from nunicrous handbooks.

T h e solid curvcs in Fig. 11 reprcscnting Pnstcs 1 and 3 havc bcen

dcrived from thc k,* vcrsus T function with thc aid of thc E, versus T

curves (for Paste 3 see Fig. 5 ) . Aggregates

Since the thermal conductivity of the cement pastc is not subjcct to large variations, it is the thermal conductivity of the aggregates that primarily tlctermines the insulating quality of thc concrcte. Unfortu- nately, data concerning thc concluctivity of aggregates are rarely at hancl and are difficult to procurc cxp~rimcntally.'~ It is, thercfore, often unavoidable to asscss this property of the aggregates from a thorough inspection and from some mineralogical data. The deductive procedure

put forward in this scction may bc regarded as typical of what a con-

crcte designcr may havc to follow.

Most concrete aggregates can bc regardecl as mixtures of crystalline and amorphous formations. Highly crystalline rocks exhibit relatively

high conductivities at room tcmperature and gr:ldually decreasing con-

ductivitics at incrcasing temperatures. T h e conductivity of amorphous

% method of measuring the thermal diffusivity of small stone particles has been derived recently [ 3 ] .

(27)

70 JOURNAL OF MATERIALS

rock formations, on the other hand, is low at room temperaturcs, rela- tively insensitive to thc chcmical composition, and increases slightly with an increase in the temperature 1261. Generally speaking, rocks with thermal conductivities over 0.005 cal/cms deg K (at room tempera- ture) can bc expccted to have predominantly crystalline, and those with conductivitics below 0.005 predon~inantly amorphous micro- structure.

From among the common rocks quartz seems to have the highest conductivity 1271: 0.025 cal/cms deg K parallel to the optic axis and 0.015 normal to that. The most common forin of quartz, the quartzitic sandstone, exhibits nearly the same thermal conductivity (in any direc- tion) as the quartz single crystals normal to the optic axis. Quartzitic sandstone aggregates were, thereforc, sclected to bc used in one of the four concretes which were expectcd to exhibit limiting thermal characteristics.

From among the more than 5 0 rocks examined by Birch and Clark a certain kind of anorthosite from Quebec (see Table 3 ) proved the best insulator; consequently it was selected to be used in another hypo- thetical concrete, also a limiting case in the normal weight concrete group.

The conlnlon lightweight aggregates owe their low thermal conduc- tivities partly to their ainorphous structures and partly to their high porosities. Because the propertfees of any one of these aggrcgates may vary in very wide ranges, for practical reasons one kind, an expanded shale product, was sclected to represent both the poorest and best aggregates in the lightweight group. The bulk dcnsity for the medium- sized and larger particles of this product was found to vary between 1.15 and 1.45 g/cm''. Since for lightweight aggregates the bulk density is, to some extent, a measure of thermal conductivity, it was assumed that two screenings from this expanded shale, one consisting entirely of particles of 1.45 g/cm" density (hereafter referred to as Expanded Shale A ) , and the other of particles of 1.15 g/cm3 (Expanded Shale B ) , would, respectively, represent the aggregates with the poorest and best thermal insulation characteristics.

T o accentuate the differences between these two hypothetical aggre- gates it was further assumed that the solid in Expanded Shale B pos- sessed a completely amorphous structure, while the solid in Expanded Shale A consisted partly of amorphous and partly of crystalline formations.

The thermal conductivities of glassy materials are known to be rela- tively insensitive to their chemical compositions. It seemed reasonable, therefore, to assume that the conductivity of "Dense Shale B," that is, of Expanded Shale B, when extrapolated to zero porosity, was com-

(28)

HARMATHY O N CONCRETE 71

parable to that of a glassy rock called obsidian; the thermal conductivity of "Dense Shale A" was similarly likened to that of a partly crystalline rock called diabase.

The conductivities of quartzitic sandstone, anorthosite, "Dense Shale A" (diabase), and "Dense Shale B" (obsidian), are plotted as curves 1 to 4 in Fig. 12 (based on data in Ref 27).

Based on -

-

Experimental Values -

---

Estimated Values

---

Calculated Values Y - Ln E

.\

<

- 0.0075 -

.-\

- D "

---_

- - - _ _ _

2,z _ - / - - - - 0 0025 - - -

-

- - -

-

- - - -

O O I ~ O 2b0

d o

d o

5b0 660

760 sbo 9bo

rooo

Temperature

C

FIG. 12-Thermal c o ~ ~ d ~ c c t i v i t y of four selected aggregates: I qrtnrtz (snrrrl- storre), 2 nnorthosite, 3 "Detlse Sllrrle A" (rlinhcrse), 4 "Derrse Slrale B" (obsirlirrtl) 5 "Expnt~ded Shale A," 6 "E.rparrded Shnle B."

For reasons discussed earlier the expression for the thermal conduc- tivity of these two hypothetical expanded shales is entirely analogous to E q 46:

Here eS is approximately constant: 0.505 and 0.608, respectively (see

Table 3). In the expression of k i (Eq 3 1 ) , k , = 0 (in the absence of moisture). k , on the other hand, because of the rough pore structure of the material (L

-

0.01 cm), will amount to about

54

of kX at 1000 C.

Using the auxiliary curves 3 and 4 for k f , curves 5 and 6 (Fig. 12) were finally obtained for the thermal conductivities of Expanded Shales A and B, respectively.

(29)

72 J O U R N A L O F MATERIALS

Concrete

With n = 3 the appropriate form of Eq 25 (combined with Eq 26) for concrete (with a single kind of aggregate) is as follows:

v,(ka

+

"li,)

+

3v,li.,

I i c = li, . . . 0 , ( 1 ~ ~

+

?I<,)

+

3va1iC

where, as mentioned earlier, v, and v, can be regarded as independent of the temperature. This formula and the information presented in Figs. 11 and 12 and in Table 4 were used to calculate the k versus temperature curves (for = 0) in Fig. 13.

I I I I I I I I I E x p e r i m e n t a l V a l u e s

'1

various g r a v e l c o n c r e t e s - v E x p a n d e d s l a g c o n c r e t e s

i}

E x p a n d e d s h a l e c o n c r e t e s - 0 P u m i c e c o n c r e t e s A A - v v - A A A A v - - a

f 3

-tTf+-"

b O I ; ! - '---4 1 I I I I I I I I 0 100 200 300 400 500 600 700 800 900 1000 Temperature C

FIG 13-Tlrrt~~rrrtl coilr/~tcii~,ii.v o f ~,nrio~t.s corto-eics. Full lirrc.r: Colrcreies 1 to 4 ill or.et~-r1r.y cotrcliliot~.

The results of numerous thermal conductivity measurenlents per- formed in this laboratory arc also plotted in Fig. 13. As expected, Concretes 1 and 2 can be regarded as "limiting" concretes in the normal weight group.; The thermal conductivities of a number of unsanded, lightweight concretes are also seen to fall between those for Concretes 3 and 4, although some have thermal conductivities slightly higher than Concrete 3. This slight discrepancy is undoubtedly due to the crudeness of the assumptions used.

' S o m e very high thermal conductivity values reported in Ref I are probably attributable to the fact that the specimens were completely s a t ~ ~ r a t e d with water- during the tests.

(30)

HARMATHY O N CONCRETE 73

I n calculating the k versus T curves for p = 0.04 and 0.08 (see Figs. 4 and 5 of Ref 2, the assu~nptions described in connection with the density of concrete were used to calculate k v . (Naturally, in the 0 to 120 C rangc the I<,, contribution to I<., is negligibly small.) T h e desorption isotherm, that is, the (pip,,.),,, versus rp relation (to be used in Eq 34) was obtained experimentally. T h e Q versus p relation was estimated as described in Ref 11. v, in turn, was taken from the

[,,

versus T plot in Fig. 3.

Naturally, when the aggregates are porous, p / p , , must be the same in the pores of both the cement paste and the aggrcgatcs; therefore, according to Eqs 34 and 31, k p and I<: must also be the same (for k,;

-

0 ) . Because of this fact, and because the aggregate porcs are not obstructed by any considerable amount of moisture (for p / p , ,

<

1 ), the increase due to moisture of the thermal conductivity of lightweight concretes is very substantial, and increases with the volume fraction of the aggregate.

The practice of taking the presence of moisture into account by attaching a k p term to I<: is strictly justifiable only under steady state conditions and at relatively high moisture contents. Nevertheless, extending to general cases the validity of this practice is hardly avoid- able, unless one is ready to sacrifice the simplicity of treating the con- duction of heat in nioist inaterials as strictly a heat flow problem, and is willing to consider the problem in its real complexity, that is, as one concerned with simultaneous heat and mass transfer [ 2 8 ] .

References

[ l l Boulder Canyon Project Final Reports, Part VII. Cement and Concrete Investigations. Bulletin 1. "Thermal Properties of Concrete," U. S. Deparl- ment of Interior, Bureau of lieclamation, Denver, Colorado, 1940.

121 Harmathy, T. Z., "Effect of Moisture on the Fire Endurance of Building Elements." Moistrrre irr Mrrret.ic11.s it! Relnriotl re1 Firc Tesr,s, A S T M STP

385, American Society lor Testing and Materials, 1965, pp. 74-94.

[ 3 ] Kalousek, G. L. and PI-ebus, A. F., "Crystal Chemistry of Hydrous C a l c i ~ ~ n l Silicates, Part Ill," Jorirtrol, American Ceramic Society, JACTA, Vol. 41,

1958, pp. 124-132.

[ 4 ] Br~lnauer, S. and Greenberg, S. A., "The Hydration of Tricalcium Silicate and p-Dicalcium Silicate at Room Temperature," Che,t~~i.srr.y of Cctrlorr, Procccditrgs Fo~trrlr ltrrertrc~riotrul Sytr~po.virrt?~, National Bureau of Standards.

Monograph 43, Vol. I, Washington, D . C., 1960, pp. 335-165.

[ 5 ] Taylor, H. F . W., "The Chemistry of Cements,'' Re.secrrclr Applicrl itr l~rrlrr.r~ry, April 1961, p p . 154-158.

[61 Powers, T. C., "The Nature of Concrete," Sj~ttrposirrt?l o ~ r Si~qtrificn~rcr of Tesrs of Coticrerc, A S T M STP 169-A, sccotrd c,di!iotr, .American Society for Testing and Materials, 1966, pp. 61-72.

[ 7 ] Copeland, L. E . and Hayes, J. C., "The Determination of Non-Evaporable U a t e r in Hardened Portland Cement Paste," A S T M Brrllcritr, Amer-icnn Society for Testing and Materials, ASTBA, No. 194, Dec. 1953, pp. 70-74.

[ 8 ] Kantro, D . L., and Copeland, L. E., "The Stoichiometry of the Hydration of Portland Cement," C l ~ e t n i . ~ [ r y of Cerr~erir, Procceclirrgs F o l i r ~ l r Ircrcrrtn-

(31)

74 JOURNAL OF MATERIALS

tiotlal Sytnposiur71, National Bureau of Standards, Monograph 43, Vol. 1, Washington, D. C., 1960, pp. 430-443.

[ 9 ] Taplin, J. H., "A Method for Following the Hydration Reaction in Portland Cement Paste," Alrstrrtlintl Jourtlal of Applied Scietice, AJACA, Vol. 10,

1959, pp. 329-345.

[ l o ] Powers, T. C., "Physical Properties of Cement Paste," Cllemistry o f Cemetit.

Proceedit~gs Fourth Itltet~tlrttiotlrtl S y t n p o s i ~ t t ~ , National Bureau of Standards, Monograph 43, Vol. 2, Washington, D. C.. pp. 577-609.

[ I l l Harmathy, T . Z., "Moisture Sorption of Building Materials," National Research Council of Canada, Division of Building Research, Technical Paper 242, NRC 9492, 1967.

[ I Z ] Eitel, W., Thermocl~etnical Methorls it1 Silicate Itlvestigntiotl, Rutgers University Press, New Brunswick, N. J., 1952, Paragraphs 140, 192, 194.

[ I 3 1 Babuschkin, W. I. and Mtschedlow-Petrossian, 0 . P., "Zur Thermodynamik der Reaktionen in den Systemen Ca(OH),-SiO,H,O, P-CIS--H,O und C:,S-H?O unter normalen und hydrothermalen Bedingungen," Siliknt Tecllt~ik, SITKA, Vo1. 10, 1959, p. 605.

[I41 Hamilton, R. L. and Crosser, 0. K., "Therrnal Conductivity of Hetero- geneous Two-Component Systems," Itrrlustrial atld Et~,qitleeritr,q Chemistry

Fut~dametltals, American Chemical Society, Vol. 1, 1962, pp. 187- 19 1.

[ I 5 1 Gorring, R. L., "Thermal Conductivity of Heterogeneous Materials," Chem-

ical Etl,~itieerit~g Progress, Vol. 57! 1961, pp. 53-59.

[I61 Kingery, W. D., Itltrorllrctiot~ to Ceramics, Wiley, New York, 1960, pp. 486 and 501.

[I71 DeVries, D. A., "The Therrnal Conductivity of Granular Materials,"

Blrlletit~, Itlstittrt ltltertl~tiotlal dlr Froirl, Atltlexe 1952-1, pp. 115-130.

[ I S ] Lo1 >, A. L., "Therrnal Conductivity: VIII, A Theory of Thermal Con- duc !'vity of Porous Materials," loirrtlal, American Ceramic Society, Vol. 37, 1954, p. 96.

[ I 9 1 Krischer, 0. and Rohnalter, H., "W'irmeleitung und Dampfdiffusion in feuchten Giitern," VDI Forscl~irt~gslreft, FRSCA, Dusseldorf, 1940, p. 402.

[ZOI Shideler, J. J., "Lightweight Aggregate Concrete for Structural Use,"

lolrrt~ril, American Concrete Institute, Vol. 54, 1957, pp. 299-328.

[ Z l ] Bogue, R. H., T l ~ e Chet?listry of Portlntld Cemetlt, second edition, Reinhold Publishing Corp., New York. 1955, pp. 526 and 593.

[ZZI Berezin, G . I.. Kiselev, A. V. and Sinitsyn, V. A,, "Heat Capacity of the Adsorption Systems Silica Gel-Water, Benzene or n-Hexane," Rirssiatl loiir-

tral o f Physical Cl~etnistry, Vol. 37, KJPCA, 1963, pp. 167-172.

[ 2 3 ] Harmathy, T. Z., "Variable-State Methods of Measuring the Thermal Prop- erties of Solids," loirrtlal o f Applied Pliysics, Vol. 35, 1964, pp. 1190-1200.

[241 Carman, A. P. and Nelson, R. A., "The Thermal Conductivity and Dif- fusivity of Concrete," Blrlletitl 122, University of Illinois, Engineering Experiment Station, 192 1.

[ 2 5 ] Harmathy, T. Z., "Peak-Time Method for Measuring Thermal Diffusivity of Small Solid Specimens," paper accepted for publication by loitrtlnl o f

Americatl ltlstitlrte o f Chetnicnl Etlgitleers.

[261 Kingery, W. D. and McQuarrie, M. C., "Thermal Conductivity: I, Concepts of Measurement and Factors Affecting Therrnal Conductivity of Ceramic Materials," loilrtlnl, American Ceramic Society, Vol. 37, 1954, p. 67.

[271 Birch, F. and Clark, H., "Thermal Conductivity of Rocks and Its Depen- dence upon Temperature," Atnericati lourtlal o f Scietrce, AJSCA, Vol. 238, 1940, pp. 542-558.

[ 2 8 ] Harmathy, T . Z., "Simultaneous Moisture and Heat Transfer in Porous Systems with Particular Reference to Drying," ltldustrinl atld Etlgineeritlg

(32)

Figure

FIG,  1-Dcgree  alld  mite  of  co~~versiorz  of  to6r1.1norite gel  i r ~   0 1 1   idacilized  cc1~1e11t  pcrste
FIG.  2-Degree  crt~d rnrc  of  L . o I ~ ~ ~ ~ ~ : Y ~ o I I   of  C ( I / L . ~ I ~ I ~ I   /~)ldro.vidc,  it1  (111  it1enli:rd  cerrierlr  paste
FIG.  4-Dila!oruerric  nt~rl rl1er.1~7ogrnviti~eIric  wlr.ves  of  Ccnlet~r  Pas!e  2
FIG.  5-Crrlcrtltrtrrl  I'LI/IIC.Y  of  tile  t1.1ir  ~ C I I S ; ~ ~ ,   brrlk  ~ l o ~
+7

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