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Thermal Expansion Kinetics: Method to Measure Permeability ofCementitious Materials, IV. Effect of Thermal Gradients and
Viscoelasticity
J. P. Ciardullo, D. J. Sweeney, and G. W. Schererw
Department of Civil and Environmental Engineering/Princeton Materials Institute Engineering Quad. E-319, PrincetonUniversity, Princeton, NJ 08544
When a porous material that is saturated with liquid is heated,the liquid expands much more than the solid phase. If the per-meability is low, then the liquid may not be able to escape as itexpands, so it expands within the pores and causes dilatation ofthe body. In that case, by analyzing the kinetics of dilatationduring a change in temperature, it is possible to extract the per-meability. Previous papers have examined the behavior of anelastic or viscoelastic (VE) porous solid subjected to a thermalcycle slow enough to avoid internal temperature gradients. How-ever, for cementitious samples, the sample size must be largeenough that thermal gradients are likely. In this paper, we showthat the effect of the gradient can be readily incorporated intothe analysis of experimental data. For cement paste, experi-ments reveal that VE relaxation has a greater influence on theresults than the gradient in temperature.
I. Introduction
WHEN a saturated porous material is heated, the thermalexpansion of the liquid in the pores is typically much
greater than that of the solid network. If the rate of heating isslow enough, the liquid will drain out of the body as it expands,but during rapid heating the liquid does not have time to escape,so it expands within the pores and forces the network to stretchlike a spring. If the heating stops, the liquid will flow out of thepores until the pressure in the liquid reaches ambient, and thesolid network will retract until the stresses in the solid phase areeliminated. Therefore, the thermal expansion kinetics of a sat-urated solid depend not only on the rate of heating, but also onthe permeability and elastic modulus of the network, the com-pressibility of the liquid, and the difference between the thermalexpansion coefficients of the solid and liquid phases. The use ofthermal expansion measurements to determine permeability iscalled thermopermeametry (TPA).1 Previous articles have pre-sented a theoretical analysis of the kinetics of this process forelastic solids,1 and experimental tests of the principle for thinplates of cement paste.2 TPA is particularly convenient for usewith cementitious materials, which have low permeabilities,2,3
but cement paste is viscoelastic (VE). In such materials, there aretwo concurrent relaxation processes: hydrodynamic relaxationas the pressure equilibrates in the pores, and VE relaxation ofthe stresses in the solid phase.4,5 Using data for stress relaxationin cement paste obtained using the beam-bending technique,6,7
we can predict the expansion kinetics; the VE effect is found tobe significant for cement paste.3
The large sample size required for TPA measurements onmortar and concrete leads inevitably to thermal gradients thatshould be taken into account in the analysis of thermal expan-sion. Extending the analysis presented previously,1 we will showthat this effect is important, and can be readily incorporated intothe elastic or VE analysis of experimental data.
II. Theory
(1) Temperature in a Cylinder
The temperature inside an infinitely long cylinder can be calcu-lated if the temperature history at the exterior surface, TS(t), isknown:8
Tðu; tÞ ¼X1n¼1
2J0ðBnuÞBnJ1ðBnÞ
expð�B2nzÞ
� T0 þ B2n
Z z
0
expðB2nz
0ÞTsðz0Þdz0� � (1)
where T0 is the initial temperature, R is the radius of the cylin-der, r is the radial coordinate, u5 r/R, z5kt/R2, k is the thermaldiffusivity, J0 and J1 are Bessel functions of the first kind oforder 0 and 1, respectively, and Bn is a root of J0, J0(Bn)5 0. Inthe experiments performed in the present study, the surfacetemperature of the sample was found to be equal to that of thesurrounding bath, so we set TS equal to the bath temperature inour calculations. Applying integration by parts, Eq. (1) can beput into a form more convenient for numerical evaluation
Tðu; tÞ ¼TsðzÞ �X1n¼1
AnJ0ðBnuÞ
�Z z
0
exp½�B2nðz� z0Þ� dTs
dz0dz0
(2)
where
An ¼X1n¼1
2
BnJ1ðBnÞ(3)
and
X1n¼1
AnJ0ðBnuÞ ¼ 1 (4)
The temperature on the axis of the cylinder is
Tð0; tÞ ¼TsðzÞ �X1n¼1
An
�Z z
0
exp �B2nðz� z0Þ
� � dTs
dz0dz0
(5)
1213
JournalJ. Am. Ceram. Soc., 88 [5] 1213–1221 (2005)
DOI: 10.1111/j.1551-2916.2005.00214.x
H. M. Jennings—contributing editor
This work was supported by a grant from the National Science Foundation, Division ofCivil and Mechanical Systems, grant CMS-0070092.
wAuthor to whom correspondence should be addressed. e-mail: [email protected]
Manuscript No. 10528. Received January 5, 2004; approved October 12, 2004.
and the average temperature on the axis of the cylinder
Tðu; tÞh i ¼ 2
Z 1
0
Tðu; tÞu du (6)
is found from Eq. (2) to be
Tðu; tÞh i ¼ TsðzÞ �Z z
0
Oc2ðz� z0Þ dTs
dz0dz0 (7)
As defined in ref. 1
Oc2ðzÞ ¼
X1n¼1
4
B2n
expð�B2nzÞ (8)
The scheme for numerical evaluation of Eqs. (5) and (7) is dis-cussed in Appendix B of ref. 1.
(2) Axial Strain in an Elastic Cylinder
The axial strain, ez, in a saturated porous cylinder containinga gradient in temperature was given in Eq. (64) of ref. 1. Thatequation can be put into a more convenient form by integratingby parts:
ez ¼es þ aSDT þ lð1� bÞbZ y
0
Oc1ðy� y0Þ
� ½ez � es � aSDT � dy0 þ lZ y
0
Oc3ðx; y� y0Þ deTS
dy0dy0
(9)
where es is the spontaneous strain (e.g., from syneresis or auto-genous shrinkage), aS is the linear thermal expansion coefficientof the solid phase, DT5T�T0, b5 (11np)/[3(1�np)], and np isPoisson’s ratio for the drained porous body; the Biot coefficientis defined by 9
b ¼ 1� Kp=KS (10)
and the constant l is
l ¼ b
bþ fðKp=KL � Kp=KSÞ(11)
where Kp is the bulk modulus of the drained body and KS is thebulk modulus of the solid phase itself, KL is the bulk modulus ofthe pore liquid (or, the reciprocal of its compressibility),f5 1�r is the porosity, and r is the volume fraction of solidsin the body. For pastes with water/cement ratios from 0.4 to 0.6and ages from 1 to 30 days, the porosity varies from about 38%to 56% and the elastic modulus is approximately described byKp/KS5r2.7 Over this range, the value of the constant l falls inthe range 0.23olo0.35.
The thermal strain in Eq. (9) is
eTS¼ fðaL � aSÞðTS � T0Þ � es (12)
where aL is the linear thermal expansion coefficient of the porefluid. In the remainder of this paper, we assume that es is neg-ligible. The reduced time is y5 t/tR, where tR is the hydrody-namic relaxation time that governs the rate of equilibration ofthe pore pressure. We have defined the function
Oc3ðx; yÞ ¼
xx� 1
� �fOc
2ðyÞ � Oc2ðxyÞg (13)
where x5 tRk/R2. In the special case where x5 1,
Oc3ð1; yÞ ¼ y Oc
1ðyÞ (14)
where, as defined in ref. 1,
Oc1ðzÞ ¼
X1n¼1
4 expð�B2nzÞ (15)
The scheme for numerical evaluation of Eq. (9) is discussed inAppendix A.
If the thermal diffusivity is very high, then Oc2ðxyÞ ! 0 and
Oc3ðx; yÞ ! Oc
2ðxyÞ, so Eq. (9) reduces to the result given in ref. 1for a sample with no internal gradient:
ez ¼aSDT þ lð1� bÞbZ y
0
Oc1ðy� y0Þ ez � aSDT½ �dy0
þ lZ y
0
Oc2ðy� y0Þ deTS
dy0dy0 ð16Þ
(3) VE Cylinder with Uniform Temperature
Equation (16) applies when the solid phase is elastic, but cementpaste is linearly VE.7,10 Under a constant uniaxial strain, thestress relaxes according to
s1ðtÞs1ð0Þ
¼ cðtÞ (17)
In such cases, the analysis is most conveniently done by elim-inating time as a variable through use of the Laplace trans-form,11 defined by
L½yðu; yÞ� ¼ yðu; sÞ ¼Z 1
0
e�syyðu; yÞdy (18)
where s is called the transform parameter. We take the trans-form with respect to the reduced time y. The transformedYoung’s modulus is
Ep ¼ cðsÞEp (19)
where Ep is, as previously, the instantaneous elastic modulus.Under the assumption that np is constant, all of the other elasticmodulus obey an analogous relationship
Ep
Ep¼ Kp
Kp¼ Hp
Hp¼ cðsÞ (20)
where Hp5Kp/b is the longitudinal elastic modulus. For anelastic solid, c(t)5 1, so cðsÞ ¼ 1=s.
When the VE relaxation behavior of the solid is taken intoaccount, the solution for the axial strain becomes5
ez ¼ aSDT þZ y
0
Ocðy� y0Þ qeTS
qy0dy0 (21)
In this case, the expression for the relaxation function, Oc, can-not be written explicitly; instead, we obtain the Laplace trans-form with respect to y,
OcðsÞ ¼1� ocðk1Þ
s 1=L� ð1� bÞ sb ocðk1Þh i (22)
where L and o are defined by
L ¼ b
,fKp
KLþ ðsb� fÞKp
KSþ sb2
" #(23)
and
ocðk1Þ ¼2I1ð
ffiffiffiffiffik1
pÞffiffiffiffiffi
k1p
I0ðffiffiffiffiffik1
pÞ¼ 2J1ði
ffiffiffiffiffik1
pÞ
iffiffiffiffiffik1
pJ0ði
ffiffiffiffiffik1
pÞ
(24)
1214 Journal of the American Ceramic Society—Ciardullo et al. Vol. 88, No. 5
Here I0 and I1 are modified Bessel functions of the first kind oforder 0 and 1, respectively; J0 and J1 are Bessel functions of thefirst kind. The imaginary form of Eq. (24) is found to be morestable for numerical evaluation. The function k1 is given by
k1 ¼fHp
KLþ ðsb� fÞHp
KSþ sb2
c
" #s
m(25)
The transform of the Biot coefficient is
b ¼ 1
s� cKp
KS(26)
and
m ¼ fHp
KLþ ðb� fÞHp
KSþ b2 (27)
In the elastic case, when c ¼ 1=s, Eq. (23) reduces to Eq. (11)and k1 reduces to s.
(4) VE Cylinder with Temperature Gradient
If the porous cylinder is VE, and also contains a temperaturegradient, the axial strain can be found by a small extension ofthe analysis presented in ref. 5. The differential equation for thepore pressure is5
qPqy
þ 3Kp
bmqeTqy
þ 1� bmb
� �b2
q Ph iqy
¼ 1
u
qqu
uqPqu
� �(28)
where P is the stress in the pore liquid (which is equal in mag-nitude to the pressure, but opposite in sign) and eT is analogousto the thermal strain in Eq. (12), but is defined in terms of theinternal temperature,
eT ¼ fðaL � aSÞ½Tðu; yÞ � T0� (29)
The solution of Eq. (28) is explained in Appendix B. The result isthe same as Eq. (21), except that the Laplace transform of therelaxation function is
OcðsÞ ¼k1
k1 � s=x
� �ocðs=xÞ � ocðk1Þ
s 1=L� ð1� bÞ sb ocðk1Þh i (30)
When the thermal diffusivity is very high ðx ! 1Þ; ocðs=xÞ ! 1and Eq. (30) reduces to Eq. (21).
III. Experimental Procedure
Type III Portland cement was prepared at a 0.5 water/cementratio using a cake mixer. Repeated strikes to the mixing bowlallowed entrained air-bubbles to escape. Molds were preparedby coating metal cylinders (diameters 19, 25, and 50 mm) with athin layer of petroleum jelly, then dipping them several timesinto melted wax, and then into cold water. The wax was cutfrom the ends of the mold with a razor blade and slipped off thecylinder; one end was covered with tape and dipped again inwax. The inside of the mold was sprayed with silicone, then filledwith the paste, and the end was resealed with wax. Once thepaste hardened, the mold was cut with a razor and the samplewas transferred into lime-saturated water until it was used. Thetheory assumes that hydrodynamic relaxation occurs by flow ofpore water to the surface of the body, rather than to internal airpockets, so it is essential that the sample be completely saturat-ed. Therefore, each sample was submerged and pressurized us-ing the equipment described in ref. 5. The pressure was set at0.7 MPa, then raised by 0.7 MPa every 4 h until approximately2 MPa was reached. The samples were left to saturate at this
pressure for 48 h. At all times during the de-molding and sat-uration process, the samples were kept at room temperature.
One cement paste sample was cast into a 76 mm plastic mold,and three thermocouples were carefully inserted vertically on thecenterline of the cylinder; the tips of the thermocouples werearranged at 1/4, 1/2, and 3/4 of the height of the mold. After thesample hardened, it was de-molded and stored in lime-saturatedwater with the thermocouple leads extending out of the bath.This sample was subjected to two heating cycles at differentrates; the thermal strain was not recorded in this experiment, butthe temperature was measured from the three internal thermo-couples, as well as the temperature of the circulating bath. Thesedata were used to calculate the thermal diffusivity of the cementpaste.
All of the thermal cycles were performed using an apparatusconsisting of an electric heater surrounding a cylindrical bathabout 80 mm in diameter. Water circulated by a pump passedthrough the chamber, entering at the bottom and draining outthe top. The temperature was monitored at the bottom of thebath, but probing with external thermocouples indicated thatthere was no vertical gradient; the temperature of the surface ofthe sample was found to be equal to that of the bath. The heat-ers were controlled by a computer, so the thermal cycle could beprogrammed. The dilatation of the sample was measured with aspring-loaded gauge with a resolution of 1 mm. The gauge wasmounted in an Invar frame that held the gauge against the sam-ple. The apparatus was calibrated by measuring the expansionof three metal standards, aluminum, copper, and steel; aftersubtracting the expansion of the Invar, the handbook valueswere obtained for the expansion of all three metals.
Thermal dilatation of cement paste samples was measuredusing heating and cooling rates of about 0.11C/min. Smallersamples were subjected to larger temperature changes (B201C);the largest sample (50 mm) could only be heated by B41C, ordamaging stresses would result. During heating, the pore pres-sure creates tensile stresses in the solid phase, and to avoid mi-crocracks (which could raise the permeability) it is necessary tokeep those stresses below B1.5 MPa for cement paste. In thefollowing experiments, the heating rate was chosen so that thepressure induced strain (i.e., the difference between the totalstrain and asDT) is t2� 10�4; given the elastic moduli of oursamples, this guarantees that the tensile stresses are low enoughto avoid damage. These small strains also allow us to treat theporosity as a constant, and ensure the validity of the linear con-stitutive equations.
Two batches of paste were prepared at water/cement5 0.5.Batch I was cast into cylinders (19, 25, and 50 mm) for use inTPA measurements. Batch II was used to make a comparison ofbeam bending and TPA; this paste was cast into cylinders19 mm in diameter, and was also formed into plates (0.27 cm� 2.1 cm� 30 cm) using PVCmolds, as described in ref. 12. Theplates were subjected to pressurization following the same pro-cedure used for the cylinders. The plates were subjected to beambending using the apparatus described in ref. 7; the deflectionimposed on the plate was 139 mm.
The porosities of the pastes from Batches I and II were de-termined by measuring the weight loss at 1051C. Paste sampleswere taken from cylinders made for TPA.
IV. Results
(1) Thermal Diffusivity
The sample with a diameter of 76 mm containing thermocouplesembedded along the axis was subjected to thermal cycles withheating/cooling rates of B0.81C/min (Profile I) and B0.11C/min (Profile II). In both experiments, the three axial thermo-couples gave nearly identical readings, so there was no evidenceof an axial temperature gradient. However, there was clearly aradial gradient, even during the slower cycle, as indicated by thedifference between the bath temperature, TS and the axial tem-perature, T(0, t), shown in Fig. 1. The data for T(0, t) were fitted
May 2005 Thermal Expansion Kinetics 1215
to Eq. (5) to obtain k; the fit for Profile I is shown in Fig. 1.Profile I yielded k5 3.2� 10�7 m2/s and Profile II yieldedk5 2.8� 10�7 m2/s; for the calculations in section IV(3), weuse the average value, k5 3.0� 10�7 m2/s. This diffusivity is atthe lower end of the spectrum reported for cement paste.13 Thecharacteristic time for equilibration of the temperature in a cyl-inder with radius R is tT � R2/k. For the samples used in thepresent study, with diameters ranging from 19 to 50 mm, tT isbetween 9 and 23 h, which is longer than the duration of theTPA experiment, so radial temperature gradients are to beexpected.
(2) Beam Bending
Three saturated plates of cement paste were subjected to beambending using the apparatus and procedure described previous-ly.7,12 The porosity of these plates was determined by measuringthe weight lost upon heating at 1051C; the volume of the evap-orable water is assumed to represent the total pore volume of thepaste.7 For these samples, the porosity was found to be 0.486, sothe volume fraction of solids is r5 0.514. One of the relaxationcurves is shown in Fig. 2; the reproducibility of the resultsfor the three experiments was excellent. The data were fitto the theoretical function given in ref. 12, using the fitting rou-tine described in ref. 7; the stress relaxation function was as-sumed to have the form given in Eq. (10) of ref. 10. The fitsyielded the elastic modulus, Ep5 12.310.7 GPa, and permea-bility coefficient, D (nm2)5 1.4310.07ZL, where ZL is the vis-cosity of the pore liquid (Pa � s). We do not have independentmeasurements of the viscosity (which may be anomalous, owingto the small pore size in the paste), but the formulas always in-clude the quotient D/ZL, so we do not need to evaluate D sep-arately. The stress relaxation function, c, obtained from the fitin Fig. 2 is written as a sum of exponential terms, so that itsLaplace transform takes a simple form5; the result is then usedto evaluate Eq. (21). The average VE relaxation time is found tobe �tVE ¼ 1:7� 105 s.
(3) Thermal Strains
The thermal expansion coefficient of the pore liquid in the pastefrom Batch I was measured using a bulb dilatometer in a studyto be reported separately.14 From 151 to 401C, which exceeds therange used in the TPA experiments, the expansion of the pore
liquid was found to be
aLð�C�1Þ ¼ ½5:66þ 0:31Tð�CÞ� � 10�5 (31)
This is considerably larger than the expansion of bulk water inthis temperature range. The existence of elevated thermal ex-pansion in the pore liquid of cement paste was revealed in an-other study in which TPA and beam-bending measurementswere made on identical samples.3 The origin of the phenomenonis not clear, but measurements of the expansion of pore water inporous glasses with narrow pore size distributions show a sys-tematic increase in aL as the pore diameter decreases belowabout 15 nm.15
For the 19 and 25 mm samples, simulation of the thermalcycle used in the TPA experiment indicated that no significanttemperature gradient would exist within the sample (Ts�/TSo0.151C). Therefore, the temperature gradient was neglected, andthe shape of the TPA curve was predicted using Eq. (21), withthe elastic and VE parameters obtained from the beam-bendingexperiments. The results shown in Fig. 3 are for a 19 mm cyl-inder of paste from the same batch (Batch II) as the beam-bend-ing samples. The shape of the axial strain curve is predictedusing the independently measured properties, with the only ad-justable parameter being the linear thermal expansion coefficientof the solid phase, aS. The VE calculation using Eq. (21) fitsslightly better than the elastic curve obtained from Eq. (16), butboth are quite good; the expansion of the solid phase is alsoshown, using as5 9.1� l0�6/1C. The overshoot is about threetimes as high as the expansion expected from as alone, owing tothe pressure generated by expansion of the pore liquid; duringthe isothermal hold, the liquid drains from the pores and themeasured expansion relaxes to aSDT.
(4) TPA
In the preceding calculation, the permeability of the sample wasknown from independent measurements, so the thermal expan-sion curve could be predicted (Fig. 3). However, we would liketo be able to use TPA to determine the permeability of thesample when beam-bending data are not available. Therefore,we will now suppose that we know Ep and aL, but notD/ZL, andwill fit Eq. (9) to the data; that is, we will take account of the
−5
0
5
10
25
30
35
40
0 1 2 3 4 5 6 7 8t (h)
∆T (
°C)
Tba
th (
°C)
Fig. 1. Right ordinate: temperature measured in water bath at surfaceof sample, Tbath and the left ordinate: the difference, DT, between Tbath
temperature measured by thermocouples embedded along the axis of thesample; symbols are measured values and solid curve is the fit to Eq. (5)for Profile I, with thermal diffusivity k as the only free parameter.
0.70
0.75
0.80
0.85
0.90
0.95
1.0
10−1 100 101 102 103 104
W(t
)/W
(0)
t (s)
H
VE
Data+
Fit
Fig. 2. Relaxation of force,W, on a saturated plate of cement followingapplication of a constant deflection. The hydrodynamic relaxation (H)resulting from equilibration of pore pressure is described by the solidcurve; the viscoelastic (VE) stress relaxation is described by the dash-dotcurve. The complete fit (dashed curve) is the product of the latter two,and is seen to provide an excellent fit to the data (symbols).
1216 Journal of the American Ceramic Society—Ciardullo et al. Vol. 88, No. 5
internal temperature variation, but will ignore the viscoelastic-ity. The thermal properties of cementitious materials have beenextensively studied, and the variation in k is not large, so it ispossible to make a reasonable estimate of the thermal gradient;if greater precision is desired, it is easy to perform an experimentof the type described in section IV(1). However, it is not easy tomeasure the stress relaxation behavior of concrete, so it is im-portant to see how serious an error results from ignoring the VEbehavior in fitting TPA data.
Cylinders of paste from Batch I, ranging in diameter from 19to 48 mm in diameter were used for TPA measurements. Basedon weight loss at 1051C, these samples were found to have solidcontents of r5 0.55 (or, porosity f5 0.45). The cylinders fromthis series were too large for bending, so we do not have inde-pendent measurements of the static modulus. The result for thepaste from Batch I (Fig. 2) falls within the range of values ob-tained for the static modulus for similar pastes (Type III) madein this lab with water/cement ratios ranging from 0.4 to0.6.3,7,10,12 Following the trend of those data, as the porositydecreases from f5 0.486 (Batch II) to 0.45 (Batch I), the mod-ulus is expected to increase to Ep � 14 GPa. We will adopt thisvalue for the following calculations.
Figure 4 shows the axial strain in a cylinder from Batch I (19mm diameter), together with the fit to Eq. (9). There is a largeovershoot during heating, then the strain relaxes toward asDT asthe pore pressure equilibrates. When the temperature drops, thestrain shows an undershoot, because contraction of the poreliquid creates negative pressure that compresses the solid net-work; during the final isothermal hold, liquid is drawn in fromthe surrounding bath and the sample re-expands as the porepressure equilibrates. Figure 5 shows the curve obtained by fit-ting the data in Fig. 4 to Eqs. (21) and (30), assuming that theVE relaxation rate is the same as found for Batch II (Fig. 2).Since the VE calculation is relatively slow, the data weresmoothed with a spline and the number of points was reducedby a factor of 10, using the numerical procedure described inref. 16. At the end of the isothermal hold, the VE curve equil-ibrates at a noticeably higher strain than the elastic curve, owingto creep of the sample caused by the pore pressure during heat-ing. The agreement is very similar to that obtained from theelastic fit (Fig. 4), but the values of the parameters are different:
tR decreases by B40% and as by B10%. The ratio of thehydrodynamic relaxation time for this sample (from Fig. 5),tR5 1.7� 104 s, to the average VE relaxation time, �tVE, istR=�tVE � 0.1. For that ratio, the analysis in ref. 5 predictserrors of just this magnitude when the VE effect is ignored.The permeability is obtained from tR, so the error inD/ZL intro-duced by ignoring the VE relaxation is on the order of 40% forthis sample. This exceeds the variation in tR found in repeatedmeasurements on the same sample, which is about 720%.12
From the elastic analysis in Fig. 4, we find D/ZL5 0.86(n �m2) � (Pa � s)�1, whereas the VE fit in Fig. 5 yields D/ZL5 1.2 (n �m2) � (Pa � s)�1. These results are consistent withthe range of values obtained from beam-bending experimentsin this lab using similar pastes with water/cement ratios from0.4 to 0.6.3,7,10,12
Figure 6 shows the calculated temperature distribution insidethe cylinder with a 50 mm diameter during the initial part of theTPA cycle, indicating that there is a lag of about 6.5 min be-
0
0.5
1
1.5
2
24
26
28
30
32
34
36
0 100 200 300 400 500 600 700
εεz Meas
εεz VE
εεzE
αs ∆T
T(°C)ε z
x104
T (
°C)
t (min)
Fig. 3. Thermal expansion of saturated paste cylinder from the samebatch as the plates used in Fig. 2; the temperature rises from 251 toB351C at a rate of B0.11C/min. Using the properties obtained frombeam bending, and the independently measured value of aL, the axialstrain is predicted using Eq. (16) (ezE) and Eq. (21) (ezVE); the expansionof the solid phase is also shown, where aS5 9.1� 10�6/1C.
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
15
20
25
30
35
40
0 200 400 600 800 1000
ε zx1
04
T (
°C)
t (min)
αs ∆T (calc)
εz
T(°C)
Fig. 4. Thermal expansion of saturated paste cylinder (diameter5 19mm) from Batch I. Calculated strains (ez and aSDT) are fits of Eq. (9) tothe data (solid symbols), using the independently measured value of aL,with two free parameters whose values are found to be tR5 2.8� 104 sand aS5 10.8� l0�6/1C. The temperature cycle, T (1C), is plotted againstthe right ordinate.
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
15
20
25
30
35
40
0 200 400 600 800 1000 1200
ε z x10
4
T (
°C)
t (min)
αs ∆T (calc)
εz
T(°C)
Fig. 5. Thermal expansion of saturated paste cylinder (diameter5 19mm) from Batch I (smoothed data from Fig. 4). Strain ez was fitted toEqs. (21) and (30), using the independently measured value of aL and therelaxation function from Fig. 2. The two free parameters are found to betR5 1.7� 104 s and aS5 9.9� 10�6/1C.
May 2005 Thermal Expansion Kinetics 1217
tween the surface and centerline temperatures. Figure 7 com-pares the results of fits of the measured expansion to the elasticanalysis, taking account of the temperature distribution by usingEq. (9), or assuming uniform internal temperature by using Eq.(16). The calculated strain clearly precedes the measured strain,if the internal temperature is assuming to equal the surface tem-perature, TS, whereas the location of the peak agrees with thedata when the temperature distribution is taken into account. Ifthe distribution is ignored, but the uniform temperature is setequal to the average temperature calculated from Eq. (7), thenthe location of the peak is accurately predicted. The three fitsyield permeability values within 5% of one another, so neglect-ing the temperature distribution does not cause serious errors inthis case. In view of the shift in the peak location, it is evidentlyadvisable to take account of the temperature distribution forlarger samples, particularly since the calculation is fast.
To estimate the importance of VE relaxation for the 50 mmsample, the calculation in Fig. 8 was done using Eq. (30). The fitis quite good when using the VE relaxation function from Fig. 2,and adjusting tR to 1.67� 105 s and aS to 14.4� 10�6/1C. Whenthe second peak is expanded in Fig. 8(b), it is evident that thestrain directly follows the temperature, so that peak does notrepresent an overshoot of the type seen in Fig. 4. Rather, thehydrodynamic relaxation is so slow that the pore pressure doesnot relax significantly, and the saturated body behaves as if itwere an elastic body with a high thermal expansion coefficient;thus, that peak does not permit evaluation of the permeability.
The indication of hydrodynamic relaxation is the slow de-crease in ez between about 700 and 1500 min. During that timethe elastic simulation (not shown) relaxes toward asDT, but theVE simulation retains a higher strain, owing to creep caused bypore pressure. The curves are nearly identical if we use Eq. (21),rather than Eq. (30), and assume that the temperature is uniformand equal to /TS. This indicates that the temperature gradientis of minor importance, compared with the relaxation. Indeed,for the present sample, the characteristic time for thermal equi-libration (R2/k5 2100 s) is about two orders of magnitude faster
22
23
24
25
26
27
0 50 100 150 200
TS
<T>
Taxial
T (
°C)
t (min)
Fig. 6. Calculated temperature distribution in cylinder of paste (diam-eter 50 mm) subjected to thermal cycle shown in Fig. 8, indicating tem-perature of bath, TS5T(1, t), average temperature /TS, and centerlinetemperature, Taxial5T(0, t).
0.5
1
1.5
2
2.5
22
23
24
25
26
27
28
29
30
550 600 650
εεz(meas)εεz(T(u,t)) εεz(<T> )εεz(TS)
TS (°C)
ε z x1
04
T (
°C)
t (min)
Fig. 7. Comparison of measured strain for the 50 mm sample (symbols)with fits to the elastic analyses (left ordinate): ez(T(u, t)) is the fit to Eq.(9) (dashed), which takes account of the internal gradient; the other twocurves are fits to Eq. (16), which assumes that the internal temperature isconstant and equal to the average temperature, ez(/TS) (solid curve) orthe surface temperature, ez(TS) (dash-dot curve). Temperature historyplotted against right ordinate.
0.0
0.5
1.0
1.5
2.0
22
23
24
25
26
27
28
29
30
0 500 1000 1500
αs∆T
εz
T(°C)
s∆T
εz
T(°C)
ε zx1
04ε z
x104
T (
°C)
T (
°C)
t (min)
t (min)
0.0
0.5
1.0
1.5
2.0
2.5
22
23
24
25
26
27
28
29
30
500 550 600 650 700 750
α
(a)
(b)
Fig. 8. Thermal expansion data (symbols) for saturated paste cylinder(diameter5 49.8 mm) from Batch I. Strains are fitted to Eqs. (21) and(30), using the independently measured value of aL and the relaxationfunction from Fig. 2. The two free parameters are found to betR5 1.7� 105 s and aS5 14.4� 10�6/1C: (a) first 24 h of cycle; (b) closeview of second peaks.
1218 Journal of the American Ceramic Society—Ciardullo et al. Vol. 88, No. 5
than the time for hydrodynamic relaxation, tR, whereas the VErelaxation time is comparable: �tVE � tR.
Figure 9 shows that when the thermal strains are fitted to theVE analysis, the values of tR obtained from cylinders with di-ameters of 19, 25.3, and 49.8 mm scale as R2, as predicted by thetheory.1 If the VE effect is neglected, the elastic fit yields highervalues, because a large overshoot can be caused by creep or lowpermeability; if the equations do not allow for creep, the fitcompensates by adopting a higher value of tR (lower permea-bility). The magnitude of the overestimation of tR shown inFig. 9 is very close to that expected on the basis of the simula-tions in ref. 5.
V. Discussion
The existence of an overshoot in thermal expansion of saturatedcementitious materials has been recognized and discussed byearlier workers17–20; the same phenomenon is seen in gels.16 Thefirst quantitative analysis of the problem21 was limited to gels, inwhich the stiffness of the solid network is negligible comparedwith that of the liquid. The analysis was extended to includemore rigid materials1 and applied to plates of saturated paste,2
but when the latter experiments were performed, it was not rec-ognized that the expansion of the pore liquid in paste is anom-alously high. That fact was suggested by comparing resultsobtained from beam bending and TPA experiments performedon companion samples,3 and was subsequently proved by directdilatometric measurements on saturated paste.14 The latterstudy included paste samples from Batch I of this study, andthe measured liquid expansion values, given in Eq. (31), are usedin the present work.
The present experiments demonstrate that the thermal ex-pansion kinetics of saturated cement paste can be accuratelypredicted, if both the hydrodynamic and VE relaxation proper-ties are known. Thus, the data from Fig. 2, together with Eq.(31), permit the prediction of the thermal strain in Fig. 3. How-ever, our goal is to use TPA to measure the permeability ofconcrete without making independent measurements of the re-laxation behavior. For cement paste and other homogenousmaterials, the bending method is to be preferred,6,7 because theexperiment is easier and it provides the VE relaxation behavior,
as well as the permeability. However, that method is not prac-tical for heterogeneous materials, such as concrete, because arepresentative volume is necessarily large; therefore, it is morepractical to use TPA with a large sample (such as the cylindersconventionally used for strength testing). Larger samples presenttwo difficulties: they are big enough to contain gradients in T,and the cycles are long enough to permit significant VE relax-ation. The temperature distribution is readily taken into accountusing Eq. (9), and the thermal diffusivity of cementitious mate-rials falls into a narrow enough range, so that accurate estimatescan be made. Moreover, the simulations done in this study showthat the error resulting from neglect of the gradients is not se-vere. The more difficult problem is to take account of the VEbehavior of cement paste.
The pore pressure resulting from thermal expansion mis-match between the solid and liquid components causes signifi-cant creep of cement paste during TPA cycles. The effect isparticularly severe in the present experiments, because the sam-ples are pure paste; concrete shows less creep than paste, thanksto the reinforcing effect of the aggregate, and it has higher per-meability, owing to the high porosity of the interface betweenthe aggregate and the paste. Therefore, the error from neglect ofVE relaxation is expected to be less important for concrete thanfor the paste samples used in this work. In the present case, thelargest error in the permeability that results from neglect of VEbehavior is less than a factor of two.5 Therefore, TPA experi-ments performed on mortar or concrete should yield muchsmaller errors, even if the data are analyzed without taking ac-count of VE effects or temperature gradients.
VI. Conclusions
Given independent measurements of permeability and VE re-laxation behavior, the thermal expansion kinetics of a saturatedporous body can be precisely predicted. Conversely, given inde-pendent measurement of the elastic modulus, the thermal ex-pansion kinetics can be analyzed to determine the permeability.This method, TPA, is a relatively rapid method for measuringthe permeability of saturated porous materials. This work showsthat both temperature gradients and VE relaxation affect theresults, but the errors resulting from neglect of those effects areless than a factor of two in the worst case (viz., a large sample ofpure cement paste). Therefore, the permeability can be estimatedwith reasonable accuracy by fitting thermal expansion kineticsto the elastic analysis, Eq. (9) (if the thermal diffusivity can beestimated) or Eq. (16). The error from neglect of VE relaxationis more important. As noted in ref. 5, if VE strain is significant,then the value of tR found from TPA will depart from the ex-pected R2 dependence, as in Fig. 9.
Appendix A: Evaluating Integrals
To evaluate Eq. (9), define
g1ðtÞ ¼ ezðtÞ � aS DTðtÞh i (A-1)
g2ðtÞ ¼ g3ðtÞ ¼ tRdeTdt
(A-2)
The integrals to be evaluated have the following form:
Hð1Þn ¼
Z t
0
exp �B2nðt� t0ÞtR
� �g1ðt0ÞtR
dt0 (A-3)
Hð2Þn ¼
Z t
0
exp �B2nðt� t0ÞtR
� �g2ðt0ÞtR
dt0 (A-4)
Hð3Þn ¼
Z t
0
exp � xB2nðt� t0ÞtR
� �g3ðt0ÞtR
dt0 (A-5)
0
1 105
2 105
3 105
4 105
5 105
0 1 2 3 4 5 6 7
VEElastic
τ R (s
)
R2 (cm2)
Fig. 9. Hydrodynamic relaxation time, tR, versus the square of the ra-dius of the cylinder, for paste samples from Batch I with radii of 19, 25.3,and 49.8 mm; triangles represent fits to the viscoelastic (VE) analysis,Eqs. (21) and (30), and circles represent fits to elastic analysis, Eq. (9).The dashed line represents the expected dependence of tR on R2.
May 2005 Thermal Expansion Kinetics 1219
Following the method presented in Appendix B of ref. 1, thecurrent value of each integral can be obtained from the previousvalue:
HðmÞn ðtþ DtÞ ¼HðmÞ
n ðtÞdmn þgmðtþ DtÞ
k2mn
1� 1� dmn
k2mnDy
� �
� gmðtÞk2mn
dmn �1� dmn
k2mnDy
� �(A-6)
where Dy ¼ Dt=tR; k21n ¼ k22n ¼ B2n and k
23n ¼ xB2
n, and
dmn ¼ expð�k2mnDt=tRÞ (A-7)
The current value of the strain is obtained from
g1ðtþ DtÞ ¼fC1ðtÞ þ C2ðtÞþ ½C3ðtÞ � C4ðtÞ�x=ðx� 1Þg=C5
(A-8)
where
C1ðtÞ ¼ 4ð1� bÞX1n¼1
Hð1Þn ðtÞd1n (A-9)
C2ðtÞ ¼ð1� bÞg1ðtÞX1n¼1
4
B4n
1� d1nDy
�X1n¼1
4d1nB2n
" #(A-10)
C3ðtÞ ¼X1n¼1
4
B2n
Hð2Þn ðtÞ (A-11)
C4ðtÞ ¼X1n¼1
4
B2n
Hð3Þn ðtÞ (A-12)
C5 ¼1
l� ð1� bÞ 1�
X1n¼1
4
B4n
1� d1nDy
!(A-13)
Appendix B: Solution of Eq. (28)
The solution is obtained by applying the Laplace transform11
with respect to y, defined in Eq. (18). The transform of Eq. (2) is
Tðu; sÞ ¼ TsðsÞ �X1n¼1
An J0 ðBnuÞsTs � T0
sþ xB2n
!(B-1)
The transform of the derivative of T is
L qTqy
� �¼ sTðu; sÞ � T0
¼ ðsTs � T0ÞX1n¼1
AnJ0ðBnuÞxB2n
sþ xB2n
(B-2)
Therefore, the transform of Eq. (28) is
1
u
qqu
uqPqu
!� k1P ¼ k3 P
� þ k4 �
X1n¼1
cnJ0ðBnuÞ (B-3)
where
cn ¼sAnk4
sþ xB2n
(B-4)
k4 ¼3Hp
m
� �eTS
(B-5)
k3 ¼1� bmb
� �s2b2
c(B-6)
k2 ¼3Hp
m
� �eT ¼ k4 �
X1n¼1
cnJ0ðBnuÞ (B-7)
and k1 is defined in Eq. (25).If we assume that the solution to Eq. (B-3) can be written in
the form
Pðu; sÞ ¼X1n¼1
pnðsÞJ0ðBnuÞ (B-8)
then the transform of the average pressure is
P�
¼ 2
Z 1
0
Pðu; sÞu du ¼X1n¼1
2J1ðBnÞpnðsÞBn
(B-9)
Substituting these results into Eq. (B-3) leads to
X1n¼1
cn � ðk1 þ B2nÞpnðsÞ
� �J0ðBnuÞ ¼ k3 P
� þ k4 (B-10)
or
X1n¼1
cn � ðk1 þ B2nÞPnðsÞ
k3 P�
þ k4
" #J0ðBnuÞ ¼ 1 (B-11)
Comparing this result with Eq. (4) reveals that the quantity inthe square bracket is equal to An, so
pnðsÞ ¼cn � An k3 P
� þ k4
�k1 þ B2
n
(B-12)
Substituting this result into Eq. (B-9) and solving for P�
weobtain
P�
¼ �X1
n¼1
4k4
ðk1 þ B2nÞðs=xþ B2
nÞ
� ��1þ k3
X1n¼1
4
B2nðk1 þ B2
nÞ
� �
(B-13)
It was shown in Appendix 2 of ref. 22 that
X1n¼1
1
xþ B2n
¼ I1ðffiffiffix
pÞ
2ffiffiffix
pI0ð
ffiffiffix
pÞ ¼ ocðxÞ (B-14)
where I0 and I1 are modified Bessel functions of the first kindof order 0 and 1, respectively. Therefore, Eq. (B-13) can bewritten as
P�
¼ k1k4
ðk1 � s=xÞocðk1Þ � ocðs=xÞk1 þ k3 � k3ocðk1Þ
� �(B-15)
The axial strain of the cylinder is1
ez ¼ aSDT � Ph i b
3Kp
� �(B-16)
Using Eq. (B-15) in this expression leads to Eq. (30).
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